Adiabatic Process

Question

Within an insulated cylinder a diatomic gas at 300 K is compressed adiabatically until its volume becomes one-fourth of the original. Using $\gamma=1.4$, what is the final temperature of the gas?

RankGuru · Accepted Answer

An adiabatic process exchanges no heat, and for an ideal gas it obeys $TV^{\gamma-1} = 	ext{constant}$, so $T_f = T_i\left(\dfrac{V_i}{V_f}ight)^{\gamma-1}$. Here the gas is compressed, so $V_i/V_f = 4$ and $\gamma - 1 = 0.4$. Then $T_f = 300	imes 4^{0.4}$. Evaluating $4^{0.4} = e^{0.4\ln 4} = e^{0.4(1.386)} = e^{0.554} \approx 1.74$, giving $T_f \approx 300	imes1.74 = 522	ext{ K}$. The value 418 K uses an exponent near 0.25, mismatching $\gamma$. The value 600 K assumes temperature scales linearly with volume ratio, ignoring the exponent. The value 300 K wrongly treats the process as isothermal, but no heat leaves the insulated cylinder so the temperature must change. The relation $TV^{\gamma-1}=	ext{constant}$ follows from combining the adiabatic law $PV^{\gamma}=	ext{constant}$ with the ideal-gas equation $PV=nRT$ to eliminate pressure. An equivalent route uses the First Law with $Q=0$, so the work done on the gas equals its rise in internal energy $nC_V\,\Delta T$, which again forces the temperature upward. As a physical check, adiabatic compression always heats a gas because work done on it raises its internal energy, so the final temperature exceeding 300 K is exactly what we expect.

Adiabatic Process

Select the correct option:

🔒 Solution Hidden from View

More thermodynamics Practice Questions

A monatomic ideal gas with gamma equal to 5/3 expands adiabatically until its volume becomes 8 times...

A gas confined in a perfectly insulated cylinder is suddenly compressed by a fast-moving piston, so ...

Compressed suddenly inside an insulated pump, a monatomic gas has work of 150 J performed on it whil...

For adiabatic compression of ideal gas, which remains constant?

A balloon suddenly bursts and the compressed gas inside rushes out into the open atmosphere very rap...

A gas is taken around a complete thermodynamic cycle and returns to its original state after several...

On a pressure-volume diagram an isothermal curve and an adiabatic curve pass through the same point ...

Two moles of an ideal gas expand isothermally at 300 K from a volume V to a volume 2V in a slow reve...

Two identical samples of the same ideal gas are kept at the same temperature but stored at very diff...

For a diatomic ideal gas the molar specific heat at constant volume is 5R/2, so what is the value of...

A heat pump uses 200 J of work to deliver a total of 800 J of heat into a warm room during each oper...

When 2 moles of a monatomic ideal gas are heated at constant pressure causing a temperature rise of ...

About This Question

Solution