Why Gas Syringes Can Miss the Start of a Fast Reaction

You mix magnesium with hydrochloric acid, start the stopwatch, and reach for the rubber stopper. Bubbles are already forming. By the time the stopper is fitted and the gas syringe is ready, several seconds have passed.

The syringe starts at 0 mL. But does that mean no hydrogen was produced before the first reading?

No. A gas syringe measures gas collected, not necessarily all the gas produced. If a fast reaction starts before the apparatus is sealed, some gas can escape. Even in a sealed apparatus, pressure changes, plunger friction, and delayed gas transfer can make the syringe respond later than the reaction.

The result is a missing or distorted early section of the gas-volume-time graph. That can make a fast reaction appear slower at the start, or even appear to accelerate when it was actually slowing down.

01The experiment starts before the syringe does

Consider the reaction:

\[ \ce{Mg(s) + 2HCl(aq) -> MgCl2(aq) + H2(g)} \]

Magnesium reacts with hydrochloric acid to produce hydrogen gas.

To measure the gas, a typical apparatus contains:

1. A conical flask holding the magnesium and acid.
2. A rubber stopper sealing the flask.
3. A delivery tube connecting the stopper to a gas syringe.
4. A graduated syringe barrel and movable plunger that indicate the collected gas volume.

The hydrogen forms in the liquid, bubbles into the gas space above it, and contributes to the gas inside the connected apparatus. As gas accumulates, it can push the syringe plunger outwards.

The syringe usually contains some air already. Hydrogen does not need to physically replace every bit of air in the tubing before the plunger can move.

What should happen if the apparatus works perfectly?

Suppose the acid touches the magnesium at exactly \(t=0\).

If the apparatus is already sealed, gas is collected completely, temperature and pressure remain constant, and the syringe responds without delay, its volume reading should track the hydrogen produced.

For a reaction that is fastest initially and gradually slows, we would predict:

• A steep initial rise in gas volume.
• A progressively smaller gradient as the reaction slows.
• A plateau once measurable gas production stops.

Now change just one detail: the stopper is not fitted until four seconds after the reaction begins.

Would the gas-volume graph simply start four seconds later?

Not quite. Gas produced before sealing may have escaped permanently. The later readings can therefore be smaller even if the chemical reaction proceeds at exactly the same rate.

An illustrative experiment: four seconds lost

The following data are invented to demonstrate the measurement problem. They are not results from a real experiment.

Imagine two otherwise identical reactions. In the first, gas is collected perfectly from the beginning. In the second, the stopper is fitted at \(t=4\ \mathrm{s}\), after 25 mL of hydrogen has escaped.

Assume all volumes refer to the same temperature and pressure, the reaction progresses identically in both cases, and no gas is lost after sealing.

By four seconds, the reaction has already produced 25 mL of hydrogen. Yet the late-connected syringe reads zero.

At ten seconds, the reaction has produced 47 mL, but the syringe has collected only 22 mL.

Neither reading means the reaction produced less hydrogen. In this model, the first 25 mL simply escaped before collection began.

Here is the important relationship.

For this particular model, after the stopper is fitted:

\[ V_{\mathrm{col}}(t)=V_{\mathrm{prod}}(t)-25 \]

where:

• \(V_{\mathrm{col}}\) is the volume collected by the syringe, in mL.
• \(V_{\mathrm{prod}}\) is the total hydrogen produced, expressed as a gas volume at the same temperature and pressure, in mL.
• \(t\) is time in seconds, measured from when the reactants first touched.

This equation applies only from \(t=4\ \mathrm{s}\) onwards, under the stated assumption that all subsequently produced gas is collected.

The missing 25 mL creates an offset. It does not automatically make every later gradient incorrect.

That distinction is worth understanding before calculating any rates.

02A flat beginning can mean two different things

There are two particularly important explanations for a delayed syringe reading. They can produce similar-looking graphs, but they describe different physical events.

Case 1: The gas escaped before collection began

The reaction starts in an open flask. Hydrogen forms, rises through the solution, and escapes into the surrounding air.

When the stopper is finally fitted, the syringe begins collecting newly produced hydrogen.

Gas that escaped cannot subsequently be collected.

If nothing else changes, the final collected volume will be lower than it would have been with complete collection. Later volume increases can still accurately represent later production.

Case 2: The apparatus was sealed, but the plunger did not move immediately

Now imagine that the flask was sealed before the reaction started.

A small amount of hydrogen forms, but the syringe plunger initially resists movement because of static friction. Gas accumulates inside the flask and tubing, increasing their pressure.

Once the pressure is sufficient to overcome the resistance, the plunger moves.

A student might see:

• No plunger movement for the first few seconds.
• A sudden increase in the syringe reading.
• A slower, steadier increase afterwards.

Does the sudden movement prove that the reaction accelerated?

No. Some of the movement may be the release of gas that had already accumulated under increased pressure.

The syringe is catching up with earlier gas production.

There is a useful analogy here: a gas syringe resembles a distance counter.

The total collected volume tells you how much gas has accumulated in the measuring device, while the change in volume per second tells you how quickly it is being collected.

If the counter starts late, the earliest part of the journey is missing. If the counter temporarily sticks, it can move rapidly later without the underlying motion speeding up.

The analogy has a limit. Gas is compressible, can dissolve in liquids, and changes volume with temperature and pressure. A syringe is not a perfect counter of gas molecules.

How to distinguish the measurement problems

The key diagnostic question is not simply whether the graph starts at zero.

It is whether the measuring apparatus was recording the reaction faithfully from the moment the reactants made contact.

03Read a rate only from the interval you actually measured

A gas-volume-time graph contains two different kinds of rate information.

The average rate over an interval is the change in measured gas volume divided by the elapsed time:

\[ \overline{r}_V=\frac{\Delta V}{\Delta t} \]

Here, \(\overline{r}_V\) is the average gas-volume rate in \(\mathrm{mL\,s^{-1}}\), \(\Delta V\) is the volume change in mL, and \(\Delta t\) is the time interval in seconds.

For a reliably collected gaseous product, the gradient is positive while its measured volume increases.

An instantaneous rate describes the rate at one particular moment. On a sufficiently reliable, smooth volume-time curve, it is found using the gradient of a tangent at that time.

Two measured points give an interval average. They do not give an exact instantaneous rate.

There is an additional problem if the beginning of the graph is missing: a tangent drawn by extending the later curve backwards is an extrapolation, not a measurement of the initial rate.

04Use another measurement to find evidence of missing gas

A useful way to investigate a suspicious early gas-volume reading is to monitor a reactant as well as the product.

For example, in the magnesium-hydrochloric acid reaction, hydrochloric acid is consumed as hydrogen forms.

The balanced equation tells us:

\[ \ce{Mg(s) + 2HCl(aq) -> MgCl2(aq) + H2(g)} \]

For every two moles of hydrochloric acid consumed, one mole of hydrogen forms.

If we know how much the acid concentration decreased, and the solution volume remained constant, we can work out how much hydrogen the chemistry produced even when the syringe failed to collect it.

There is one crucial distinction in the signs:

• Gas product volume increases, so its measured gradient is positive.
• Reactant concentration decreases, so its concentration-time gradient is negative.
• The rate of disappearance of the reactant is reported as a positive quantity by reversing that negative sign.

For example, the positive average rate of hydrochloric acid disappearance is:

\[ -\frac{\Delta[\ce{HCl}]}{\Delta t} \]

The square brackets mean concentration in \(\mathrm{mol\,L^{-1}}\).

This is a concentration disappearance rate, not a gas-volume rate. The balanced equation, solution volume, and gas conditions are needed to connect the two.

05Improve the start of the experiment, not just the graph

Suppose the original experiment mixed the reactants in an open flask and then fitted the stopper.

What needs changing?

The main problem is that the reaction begins before the measuring system is ready.

Taking syringe readings every second instead of every five seconds will not recover gas that escaped before sealing.

A revised sealed-start design

One practical approach is to keep the reactants apart inside a sealed vessel until the measurement begins.

For example:

1. Put the hydrochloric acid in a conical flask and the magnesium in a small dry inner container positioned so that they initially remain separate.
2. Seal the flask and connect the delivery tube to a freely moving gas syringe.
3. Check that the connections are airtight and that the syringe can move without sticking.
4. Tilt the vessel carefully so the acid contacts the magnesium, and start the timer at that moment.
5. Record the gas volume at short, regular intervals from the start.

The entire gas-collection system is connected before hydrogen production begins.

A successful revision should reduce gas loss during setup and make the early readings more representative of production.

However, sealing the flask beforehand only solves the gas-escape problem. It does not automatically remove syringe friction or pressure changes.

For a very fast reaction, a pressure sensor may help detect gas accumulation before the syringe plunger moves. A suitable recording system can also reduce the delay between the start of the reaction and the first measurement.

How would you test whether the revision worked?

Compare both the early measurements and the final collected volume.

If early gas previously escaped, an improved seal should generally increase the collected yield under otherwise identical conditions.

If a plunger previously stuck, the final yield might already have been close to the expected value. The improvement would instead appear in the timing and smoothness of the volume readings.

You should also check that:

• The same reactant amounts, concentrations, temperature, and mixing conditions are used.
• The syringe has sufficient capacity and does not reach its end stop.
• The collected gas is compared at consistent temperature and pressure.
• There are no leaks or significant gas losses through dissolution.
• The timing of reactant contact is defined consistently.

A useful alternative is monitoring mass loss when a gas escapes from an open reaction vessel. The balance can record the decrease in mass as gas leaves, provided evaporation, splashing, and other mass losses are controlled.

But a completely sealed flask and syringe assembly does not lose mass simply because gas forms inside it. The gas is still part of the weighed system.

Neither method automatically gives an instantaneous initial rate. That depends on whether the measurement begins promptly and responds faithfully enough to resolve the early change.

06Questions and solutions

All data in the following questions are original illustrative data. They are designed to test interpretation and experimental reasoning, not to represent actual experimental results.

Question 1

A gas syringe is connected and sealed before the reaction between magnesium and hydrochloric acid begins.

The following readings are obtained:

Assume complete gas collection at constant temperature and pressure.

Calculate the average hydrogen production rate between 5 and 15 seconds. Explain whether this value is the instantaneous rate at 15 seconds.

Solution 1

The average hydrogen production rate between 5 and 15 seconds is \(1.60\ \mathrm{mL\,s^{-1}}\). It is not the instantaneous rate at 15 seconds.

Average rate is calculated using the change in gas volume divided by the elapsed time:

\[ \overline{r}_V=\frac{\Delta V}{\Delta t} \]

Substitute the readings:

\[ \begin{aligned} \overline{r}_{5-15} &=\frac{28-12}{15-5}\\ &=\frac{16}{10}\\ &=1.60\ \mathrm{mL\,s^{-1}} \end{aligned} \]

This means that, on average, 1.60 mL of hydrogen was produced per second during that ten-second interval.

The instantaneous rate at 15 s would be found from a tangent to a reliable, smooth volume-time curve at that point.

The tempting mistake is treating a ten-second average as the rate at the end of the interval. The reaction may have slowed throughout those ten seconds, so the instantaneous rate at 15 s could be lower than 1.60 mL/s.

Question 2

Two identical magnesium-hydrochloric acid reactions begin at \(t=0\).

In Experiment A, the gas syringe is sealed and connected before the reactants touch.

In Experiment B, the reaction vessel remains open until \(t=4\ \mathrm{s}\). The stopper is then fitted, and the syringe begins recording from zero.

Both reactions have the same temperature, pressure, reactant quantities, and chemical progress. Assume all gas produced after sealing is collected, and all gas produced earlier in Experiment B escapes.

The results are:

A student argues that Experiment B must have reacted more slowly because its gas-volume readings and final volume are lower.

Evaluate this claim.

In your response, calculate the average collection rates between 8 and 12 s, determine the volume lost before Experiment B was sealed, and calculate its actual average hydrogen production rate over the first 12 seconds. Explain why none of these averages gives an exact instantaneous initial rate.

Solution 2

Experiment B did not react more slowly under the stated assumptions. Its lower readings result from 18 mL of hydrogen escaping before the apparatus was sealed.

First, compare the rates between 8 and 12 s.

For Experiment A:

\[ \begin{aligned} \overline{r}_A &=\frac{42-32}{12-8}\\ &=2.50\ \mathrm{mL\,s^{-1}} \end{aligned} \]

For Experiment B:

\[ \begin{aligned} \overline{r}_B &=\frac{24-14}{12-8}\\ &=2.50\ \mathrm{mL\,s^{-1}} \end{aligned} \]

The later average collection rates are identical.

Next, calculate the missing early volume.

Experiment A collected 18 mL in the first four seconds. Because the reactions have identical progress and Experiment B lost all gas formed before sealing, Experiment B also produced 18 mL during those seconds.

Its final collected volume is consequently 18 mL lower:

\[ 52-34=18\ \mathrm{mL} \]

The constant difference is consistent with the stated early-loss model.

By 12 s, Experiment B has produced:

\[ V_{\mathrm{prod}}=18+24=42\ \mathrm{mL} \]

Its actual first-12-second average production rate was therefore:

\[ \begin{aligned} \overline{r}_{0-12} &=\frac{42}{12}\\ &=3.50\ \mathrm{mL\,s^{-1}} \end{aligned} \]

A tempting calculation would be \(24/12=2.00\ \mathrm{mL\,s^{-1}}\). But this divides gas collected after four seconds by time measured from the beginning of the reaction.

The experiment supports equal reaction progress because the question explicitly establishes identical chemistry, and the readings are consistent with that assumption. A constant difference between two graphs, by itself, would not prove that the only problem was early escape; a calibration offset could also produce a constant difference.

Finally, even Experiment A’s early average:

\[ \frac{18}{4}=4.50\ \mathrm{mL\,s^{-1}} \]

is not an exact instantaneous rate at \(t=0\). The reaction might have been faster at first and slower towards the end of the four-second interval.

Question 3

A student measures the reaction of magnesium with hydrochloric acid using a gas syringe.

The reaction begins at \(t=0\), but the stopper is only fitted at \(t=4\ \mathrm{s}\).

Independent measurements of hydrochloric acid concentration are available. The solution volume is constant at \(50.0\ \mathrm{mL}\).

Assume that only the magnesium reaction consumes hydrochloric acid, the concentration measurements are accurate, and gas volumes are compared at \(25^\circ\mathrm{C}\) and 100 kPa.

Use \(V_m=24.8\ \mathrm{L\,mol^{-1}}\).

The student proposes that the only measurement problem was hydrogen escaping before the stopper was fitted. Once fitted, they claim, the syringe collected all additional hydrogen immediately.

Test this explanation quantitatively for the intervals 4-12 s and 12-20 s.

State what the data establish and what additional measurement could help identify the cause of any discrepancy.

Solution 3

The explanation is inconsistent with the data. Hydrogen escaping before four seconds cannot, by itself, explain why the syringe also records less gas than the chemistry produces during later intervals.

The reaction is:

\[ \ce{Mg(s) + 2HCl(aq) -> MgCl2(aq) + H2(g)} \]

Two moles of hydrochloric acid are consumed for each mole of hydrogen produced.

First, consider 4-12 s.

The acid concentration decreases by:

\[ 0.580-0.550=0.030\ \mathrm{mol\,L^{-1}} \]

The number of moles consumed is:

\[ \begin{aligned} n(\ce{HCl}) &=0.030(0.0500)\\ &=1.50\times10^{-3}\ \mathrm{mol} \end{aligned} \]

Therefore:

\[ n(\ce{H2})=7.50\times10^{-4}\ \mathrm{mol} \]

At the stated temperature and pressure:

\[ \begin{aligned} V(\ce{H2}) &=(7.50\times10^{-4})(24.8)\\ &=0.0186\ \mathrm{L}\\ &=18.6\ \mathrm{mL} \end{aligned} \]

But the syringe only increased by 14.0 mL.

The difference is:

\[ 18.6-14.0=4.6\ \mathrm{mL} \]

The syringe has already failed to account for 4.6 mL of equivalent hydrogen volume after sealing.

Now examine 12-20 s.

The concentration decrease is:

\[ 0.550-0.536=0.014\ \mathrm{mol\,L^{-1}} \]

The amount of hydrogen produced is:

\[ \begin{aligned} n(\ce{H2}) &=\frac{0.014(0.0500)}{2}\\ &=3.50\times10^{-4}\ \mathrm{mol} \end{aligned} \]

The equivalent gas volume is:

\[ \begin{aligned} V(\ce{H2}) &=(3.50\times10^{-4})(24.8)\\ &=8.68\ \mathrm{mL} \end{aligned} \]

The syringe increase is only:

\[ 20.5-14.0=6.5\ \mathrm{mL} \]

The discrepancy in this interval is therefore:

\[ 8.68-6.5=2.18\ \mathrm{mL} \]

These two discrepancies occur after the stopper was fitted. Early escape alone cannot explain them.

Possible additional problems include continued leakage, gas temporarily accumulating under pressure instead of moving the plunger, or hydrogen remaining in the liquid or elsewhere in the apparatus rather than contributing immediately to the syringe reading.

The data do not uniquely identify which explanation is correct.

A pressure sensor in the sealed system, together with a later final-volume measurement after pressure has stabilised, would help distinguish temporary accumulation from permanent loss. Checking the connections for leaks would also be useful.

The important constraint is the independently measured acid consumption. Under the assumptions given, that consumption requires more hydrogen formation during both later intervals than the syringe reports.

Question 4

Two matched experiments investigate the reaction between calcium carbonate and hydrochloric acid:

\[ \ce{CaCO3(s) + 2HCl(aq) -> CaCl2(aq) + H2O(l) + CO2(g)} \]

In Experiment A, the reaction takes place in a flask on an electronic balance. Carbon dioxide escapes through a cotton-wool plug, and the decreasing mass is recorded from \(t=0\).

In Experiment B, gas is collected using a syringe. The reaction begins at \(t=0\), but the syringe is not sealed to the flask until \(t=5\ \mathrm{s}\).

The results are:

Assume that both reactions follow the same chemical time course, all produced carbon dioxide promptly leaves the liquid in Experiment A, and there is negligible evaporation or splashing. Gas volumes refer to \(25^\circ\mathrm{C}\) and 100 kPa.

Use:

\[ M(\ce{CO2})=44.0\ \mathrm{g\,mol^{-1}} \]

and

\[ V_m=24.8\ \mathrm{L\,mol^{-1}} \]

Determine the equivalent volume of carbon dioxide produced during the first five seconds.

Then test whether the gas syringe records the correct additional amount of carbon dioxide during the following two five-second intervals.

Finally, explain why putting the entire sealed flask-and-syringe apparatus on a balance would not directly give the same mass-loss results as Experiment A, and why the first-five-second mass loss cannot establish the instantaneous rate at \(t=0\).

Solution 4

The first five seconds produced an equivalent 12.4 mL of carbon dioxide, and the later syringe readings agree with the mass-loss data to the stated precision. The early gas was missed by Experiment B, but the later measurements are consistent with complete collection.

First, convert mass lost to moles of carbon dioxide.

During the first five seconds, the mass decreases by 0.0220 g.

\[ \begin{aligned} n(\ce{CO2}) &=\frac{0.0220}{44.0}\\ &=5.00\times10^{-4}\ \mathrm{mol} \end{aligned} \]

Using the molar volume:

\[ \begin{aligned} V(\ce{CO2}) &=(5.00\times10^{-4})(24.8)\\ &=0.0124\ \mathrm{L}\\ &=12.4\ \mathrm{mL} \end{aligned} \]

This represents the gas produced before the syringe in Experiment B began collecting.

Now consider 5-10 s.

The additional mass lost in Experiment A is:

\[ 0.0500-0.0220=0.0280\ \mathrm{g} \]

The corresponding gas volume is:

\[ \begin{aligned} n(\ce{CO2}) &=\frac{0.0280}{44.0}\\ &=6.36\times10^{-4}\ \mathrm{mol}\\ V(\ce{CO2}) &=(6.36\times10^{-4})(24.8)\\ &=15.8\ \mathrm{mL} \end{aligned} \]

Experiment B also records 15.8 mL during this interval.

For 10-15 s, the additional mass lost is:

\[ 0.0640-0.0500=0.0140\ \mathrm{g} \]

Therefore:

\[ \begin{aligned} n(\ce{CO2}) &=\frac{0.0140}{44.0}\\ &=3.18\times10^{-4}\ \mathrm{mol}\\ V(\ce{CO2}) &=(3.18\times10^{-4})(24.8)\\ &=7.89\ \mathrm{mL} \end{aligned} \]

Experiment B’s syringe increases by:

\[ 23.7-15.8=7.9\ \mathrm{mL} \]

The results again agree to the precision of the readings.

This supports an early collection loss rather than an ongoing collection problem, under the question’s assumptions.

The next distinction concerns what the balance actually measures.

In Experiment A, gas escapes from the weighed flask. The mass of the material remaining on the balance decreases.

In a completely sealed flask-and-syringe assembly, the carbon dioxide remains inside the weighed system. The total mass of that system does not decrease just because a gas has formed or moved into the syringe.

A balance therefore cannot measure reaction progress through mass loss unless material actually leaves the weighed system.

Finally, the average mass-loss rate over the first five seconds is:

\[ \begin{aligned} \overline{r}_m &=\frac{0.0220}{5}\\ &=0.00440\ \mathrm{g\,s^{-1}} \end{aligned} \]

Here, \(\overline{r}_m\) is the positive average rate of mass loss. The gradient of the remaining-mass graph would instead be negative.

The equivalent average gas-volume production rate is:

\[ \begin{aligned} \overline{r}_V &=\frac{12.4}{5}\\ &=2.48\ \mathrm{mL\,s^{-1}} \end{aligned} \]

Neither value gives the exact rate at \(t=0\). Both describe what happened over the complete first five seconds.

The mass method can reveal early gas production that a late-connected syringe misses, but it still needs sufficiently early measurements to determine the initial gradient.

Question 5

A student designs a sealed-start magnesium-hydrochloric acid experiment to eliminate gas escape.

The flask, delivery tube, and gas syringe are connected before the reaction starts. A pressure sensor monitors the gas inside the apparatus.

At \(t=0\):

• The gas already inside the sealed apparatus occupies \(100.0\ \mathrm{mL}\), including all fixed headspace and dead volume.
• Its absolute pressure is \(100.0\ \mathrm{kPa}\).
• The syringe plunger is at its zero-displacement position.
• The gas temperature is \(25^\circ\mathrm{C}\).

The gas syringe contains a movable plunger, but it sometimes sticks before beginning to move.

Treat all gases as ideal. Assume that temperature remains constant, the pressure is uniform throughout the connected gas space, no hydrogen escapes or dissolves, and the only change in the number of gas molecules comes from hydrogen production.

The syringe displacement adds to the initial \(100.0\ \mathrm{mL}\) gas volume.

The measurements are:

A student concludes:

“The reaction produced no hydrogen in the first three seconds. Between three and six seconds, it must have been producing hydrogen at 20 ÷ 3 mL/s. Therefore, the reaction accelerated sharply before slowing down.”

Evaluate this conclusion using the ideal gas equation.

Calculate the cumulative hydrogen produced at 3, 6, and 12 s, expressing each amount as the volume it would occupy at \(25^\circ\mathrm{C}\) and 100.0 kPa.

Use these results to compare the actual average hydrogen production rates over 0-3 s, 3-6 s, and 6-12 s.

Finally, determine whether the corrected data prove that the instantaneous rate was exactly \(4.00\ \mathrm{mL\,s^{-1}}\) at \(t=0\), or that the rate decreased continuously throughout the experiment. Justify your answer by considering different possible gas-production histories consistent with the measurements.

Solution 5

The student’s conclusion is incorrect. Hydrogen was already being produced during the first three seconds, and the corrected average production rates decrease across the three measured intervals.

The apparent surge between three and six seconds includes expansion of gas that had accumulated under increased pressure while the plunger was stationary.

The central constraint is conservation of the amount of gas inside the sealed apparatus.

Step 1: Account for both gas pressure and total gas volume.

The ideal gas equation is:

\[ PV=nRT \]

Here, \(P\) is absolute pressure, \(V\) is the total gas volume, \(n\) is the total amount of gas in moles, \(R\) is the ideal gas constant, and \(T\) is temperature in kelvin.

Because temperature is constant, the total number of moles of gas is proportional to \(PV\).

The initial gas has:

\[ P_0=100.0\ \mathrm{kPa} \]

and

\[ V_0=100.0\ \mathrm{mL} \]

As hydrogen forms, it adds gas molecules to those already present.

At any later time, the total gas volume is:

\[ V_0+V_s \]

where \(V_s\) is the extra volume created by syringe displacement.

The increase in total gas amount is exactly the hydrogen produced, under the stated assumptions.

To express that increase as an equivalent volume at the original pressure, we can use:

\[ V_{\mathrm{H_2,eq}} = \frac{P}{P_0}(V_0+V_s)-V_0 \]

Here, \(V_{\mathrm{H_2,eq}}\) is the volume the newly produced hydrogen would occupy at the reference pressure and temperature.

This expression comes from comparing the initial and later values of \(PV\). It is not simply the syringe displacement.

Step 2: Calculate hydrogen production during the first three seconds.

At 3 s:

\[ P=112.0\ \mathrm{kPa} \]

and

\[ V_s=0\ \mathrm{mL} \]

Substitute:

\[ \begin{aligned} V_{\mathrm{H_2,eq}} &=\frac{112.0}{100.0}(100.0)-100.0\\ &=12.0\ \mathrm{mL} \end{aligned} \]

Therefore, 12.0 mL of equivalent hydrogen volume has been produced by 3 s.

The syringe has not moved, but the increased pressure proves that extra gas molecules are present.

Under the question’s assumptions, those extra molecules must be hydrogen from the reaction.

The average hydrogen production rate from 0 to 3 s is:

\[ \begin{aligned} \overline{r}_{0-3} &=\frac{12.0}{3}\\ &=4.00\ \mathrm{mL\,s^{-1}} \end{aligned} \]

Step 3: Calculate cumulative production at six seconds.

At 6 s:

\[ P=100.0\ \mathrm{kPa} \]

and

\[ V_s=20.0\ \mathrm{mL} \]

The total gas volume is now 120.0 mL.

Therefore:

\[ \begin{aligned} V_{\mathrm{H_2,eq}} &=\frac{100.0}{100.0}(120.0)-100.0\\ &=20.0\ \mathrm{mL} \end{aligned} \]

A total of 20.0 mL of equivalent hydrogen volume has been produced since the reaction began.

However, 12.0 mL had already been produced by 3 s.

Only:

\[ 20.0-12.0=8.0\ \mathrm{mL} \]

was produced during the interval from 3 to 6 s.

The average production rate over that interval is:

\[ \begin{aligned} \overline{r}_{3-6} &=\frac{8.0}{3}\\ &=2.67\ \mathrm{mL\,s^{-1}} \end{aligned} \]

The plunger moved through 20.0 mL during those three seconds, but only 8.0 mL of equivalent gas volume was newly produced.

The difference came from the expansion of gas that was already inside the apparatus under increased pressure.

The student’s 20 ÷ 3 calculation gives the average syringe displacement rate, not the hydrogen production rate.

Step 4: Calculate production from 6 to 12 seconds.

At 12 s, the pressure is again 100.0 kPa, and the syringe displacement is 28.0 mL.

The cumulative hydrogen production is:

\[ \begin{aligned} V_{\mathrm{H_2,eq}} &=\frac{100.0}{100.0}(128.0)-100.0\\ &=28.0\ \mathrm{mL} \end{aligned} \]

Additional production from 6 to 12 s is:

\[ 28.0-20.0=8.0\ \mathrm{mL} \]

The average rate is:

\[ \begin{aligned} \overline{r}_{6-12} &=\frac{8.0}{6}\\ &=1.33\ \mathrm{mL\,s^{-1}} \end{aligned} \]

The corrected comparison is:

The average production rate falls from one interval to the next.

The apparent acceleration seen in the syringe alone is a measurement-response effect, not evidence that hydrogen production accelerated between the first two intervals.

Step 5: Decide what the corrected measurements still cannot tell us.

The corrected first-three-second average is 4.00 mL/s.

But it does not prove that the instantaneous rate at \(t=0\) was exactly 4.00 mL/s.

Consider two possible production histories during the first three seconds.

In Model A, hydrogen forms at a constant rate of 4.00 mL/s throughout that interval.

The total production is:

\[ 4.00(3)=12.0\ \mathrm{mL} \]

In Model B, hydrogen forms at 2.00 mL/s during the first second and 5.00 mL/s during the following two seconds.

The total production is:

\[ 2.00(1)+5.00(2)=12.0\ \mathrm{mL} \]

Both models produce exactly 12.0 mL by three seconds.

Both are consistent with the recorded gas amount at that time. Yet the rates at the beginning are different, and Model B even includes an early increase.

They can also be followed by production histories that give the measured additional 8.0 mL in each later interval.

Therefore, the measured interval averages do not establish the detailed instantaneous-rate history.

The data show that the average production rate decreases across successive intervals. They do not prove that the reaction rate decreased at every instant.

This is the deeper limitation of early gas-volume measurements.

Even if we correct for pressure, collect every gas molecule, and calculate the amount produced accurately at each recorded time, measurements taken several seconds apart cannot uniquely determine everything that happened between them.

A reliable initial instantaneous rate requires sufficiently detailed measurements close to the moment of reaction, together with a measurement system that responds to gas formation without significant delay.

07The next decision: measure gas movement or chemical change?

When a fast reaction produces gas, the first question should be whether the collection system can respond quickly enough to follow the chemistry.

If gas escapes before sealing, the early quantity is lost from the record. If gas accumulates under pressure or the plunger sticks, the early quantity may be present but temporarily invisible to the syringe. If gas dissolves or temperature changes, the relationship between gas produced and syringe displacement becomes more complicated.

For a short early interval, an independent measurement of reactant concentration, gas pressure, or mass loss may be more informative than a gas syringe alone.

The next useful comparison is How to Compare Reaction Rates from Mass-Loss Graphs, where the same reaction can look different depending on what the apparatus actually measures.

The central rule is simple: calculate the gradient of the quantity that was genuinely measured, over the time it was genuinely measured, before interpreting it as the reaction rate.