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Difficulty: MediumKinetic Theory of Matter and Pressure of Gases

A sample of an ideal gas has an average translational kinetic energy of EkE_k per molecule at an initial temperature of 27∘C27^\circ\text{C}. If the gas is heated at constant volume until the average kinetic energy per molecule doubles to 2Ek2E_k, what is the final temperature of the gas in degrees Celsius?

  1. A
    54∘C54^\circ\text{C}
  2. 327∘C327^\circ\text{C}Answer
  3. C
    600∘C600^\circ\text{C}
  4. D
    927∘C927^\circ\text{C}

Answer

The final temperature of the gas is 327∘C327^\circ\text{C}.
In the kinetic theory of matter, average translational kinetic energy per molecule is directly proportional to absolute temperature (Ek∝TE_k \propto T). Initial temperature T1=27+273=300 KT_1 = 27 + 273 = 300\text{ K}. Doubling the kinetic energy doubles the Kelvin temperature to 600 K600\text{ K}. Subtracting 273273 yields 327∘C327^\circ\text{C}.

Step-by-Step Solution

1
Convert the initial temperature from Celsius to Kelvin.
T1=27∘C+273=300 KT_1 = 27^\circ\text{C} + 273 = 300\text{ K}
Kinetic theory calculations require thermodynamic temperature measured on the absolute Kelvin scale.
2
Apply the proportional relationship between average translational kinetic energy and temperature.
Since Ek∝TE_k \propto T, doubling EkE_k means T2=2×T1=2×300 K=600 KT_2 = 2 \times T_1 = 2 \times 300\text{ K} = 600\text{ K}.
The average kinetic energy of gas molecules is directly proportional to the absolute temperature.
3
Convert the final temperature from Kelvin back to degrees Celsius.
t2=600 K−273=327∘Ct_2 = 600\text{ K} - 273 = 327^\circ\text{C}
The question asks for the final temperature specifically in degrees Celsius.

Key Concept

Average translational kinetic energy of an ideal gas molecule is directly proportional to its absolute temperature (Ek=32kBTE_k = \frac{3}{2} k_B T).
Estimated Time:1m 30s
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Topics

Kinetic Theory of Matter and Pressure of Gases

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