Show Attempt History rrent Attempt in Progress The catalyzed decomposition of ethanol at $$ 327^{\circ} \mathrm{C} $$ has a rate constant of $$ 4.00 imes 10^{-5} \mathrm{~L} \mathrm{~mol} \mathrm{l}^{-1} \mathrm{~s}^{-1} $$. A graph of the reciprocal of ethanol concentration versus time gives a straight line. The chemical equation for the reaction at this temperature is If the initial concentration of ethanol is $$ 0.024 \mathrm{~mol}^{-1} $$, how long will it take for the pressure to reach 1.4 atm at $$ 327^{\circ} \mathrm{C} $$ ? i $$ \quad \mathrm{C}_{2} \mathrm{H}_{4} \mathrm{OH}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) $$ eTextbook and Media Hint

Question

Show Attempt History rrent Attempt in Progress The catalyzed decomposition of ethanol at $$ 327^{\circ} \mathrm{C} $$ has a rate constant of $$ 4.00 	imes 10^{-5} \mathrm{~L} \mathrm{~mol} \mathrm{l}^{-1} \mathrm{~s}^{-1} $$. A graph of the reciprocal of ethanol concentration versus time gives a straight line. The chemical equation for the reaction at this temperature is If the initial concentration of ethanol is $$ 0.024 \mathrm{~mol}^{-1} $$, how long will it take for the pressure to reach 1.4 atm at $$ 327^{\circ} \mathrm{C} $$ ? i $$ \quad \mathrm{C}_{2} \mathrm{H}_{4} \mathrm{OH}(\mathrm{g})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) $$ eTextbook and Media Hint

UpStudy AI · Accepted Answer

We are given that the decomposition of ethanol is second order in ethanol. (A straight‐line plot of 1/[EtOH] versus time indicates a second‐order reaction.) In addition, one mole of ethanol is converted into two moles of products according to

C₂H₅OH(g) → C₂H₄OH(g) + H₂O(g).

Because the reaction produces two gaseous products from one mole of ethanol, the total number of moles in the gas phase increases. (We assume that the reaction is carried out in a rigid container so that pressure is directly proportional to the number of moles at constant temperature.) In many such problems the initial pressure is 1.00 atm, so that we can use the proportionality to relate the change in mole number to the change in pressure.

Below are the steps that lead us to the answer.

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Step 1. Relate Pressure Change to Extent of Reaction

Let the initial concentration (and hence number of moles per liter) of ethanol be [EtOH]₀ and let x be the molar extent (in mol L⁻¹) of ethanol that has reacted at time t. Then:

• Ethanol remaining = [EtOH]₀ – x.
• Each mole of ethanol produces one mole of each product, so the amount of each product formed is x.
• Thus, the total concentration of gases at time t is:
  Total = ([EtOH]₀ – x) + x + x = [EtOH]₀ + x.

At t = 0, the system has only ethanol so that the initial pressure P₀ is due to [EtOH]₀. If we assume P₀ = 1.00 atm, then the pressure at time t is proportional to the total concentration:

P_t = (Total / [EtOH]₀) · P₀ = ([EtOH]₀ + x) / [EtOH]₀.

We are told that the pressure reaches 1.4 atm. Thus,

1.4 atm = ([EtOH]₀ + x) / [EtOH]₀         (1)

Given [EtOH]₀ = 0.024 mol L⁻¹, solve (1):

[EtOH]₀ + x = 1.4 · [EtOH]₀ = 1.4 × 0.024 = 0.0336 mol L⁻¹,
  So, x = 0.0336 – 0.024 = 0.0096 mol L⁻¹.

Thus, at the time when the pressure reaches 1.4 atm, the ethanol left is

[EtOH]_t = [EtOH]₀ – x = 0.024 – 0.0096 = 0.0144 mol L⁻¹.
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Step 2. Use the Integrated Rate Law for a Second-Order Reaction

For a second-order reaction in one reactant,
  1/[EtOH]_t = 1/[EtOH]₀ + k t,
where k is given as 4.00 × 10⁻⁵ L mol⁻¹ s⁻¹.

Plug in the numbers:

1/(0.0144) = 1/(0.024) + (4.00 × 10⁻⁵)t.

Calculate the reciprocals:

1/0.0144 ≈ 69.44 L mol⁻¹,
  1/0.024 ≈ 41.67 L mol⁻¹.

Thus,

69.44 = 41.67 + (4.00 × 10⁻⁵)t.

Solve for t:

(4.00 × 10⁻⁵)t = 69.44 – 41.67 = 27.77 L mol⁻¹,
  t = 27.77 / (4.00 × 10⁻⁵) ≈ 694,250 seconds.
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Step 3. Convert the Time to More Familiar Units

To convert seconds to hours:

t (in hours) = 694,250 s ÷ 3600 s/h ≈ 192.85 hours.

Or, in days:

t ≈ 192.85 h ÷ 24 ≈ 8.04 days.
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Answer

It will take approximately 694,250 seconds (about 193 hours or roughly 8 days) for the pressure to reach 1.4 atm at 327°C.
It will take approximately 694,250 seconds (about 193 hours or roughly 8 days) for the pressure to reach 1.4 atm at 327°C.

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