Question 576: Thermodynamics & KTG

Question

An ideal gas initially at $0^{\circ} \mathrm{C}$ temperature, is compressed suddenly to one fourth of its volume. If the ratio of specific heat at constant pressure to that at constant volume is $3 / 2$, the change in temperature due to the thermodynamic process is _________ K.

Accepted Answer

WeÃ¢ÂÂre given an ideal gas initially at $$T_i = 0^\circ \mathrm{C} = 273\,\mathrm{K}.$$ The gas is suddenly compressed to one-fourth of its initial volume, and we are told that
$$\gamma = \frac{c_p}{c_v} = \frac{3}{2}\,.$$
For an adiabatic process (one in which there is no heat exchange), if the process were reversible, the relation between the temperature and volume would be
$$T\, V^{\gamma-1} = 	ext{constant}\,.$$
Even though the compression is Ã¢ÂÂsuddenÃ¢ÂÂ (and hence irreversible), the final equilibrium state of the gas is uniquely determined by its internal energy (which is a state function). That allows us to use the adiabatic relation between the initial and final states.
Here are the steps:
Write the adiabatic relation between the initial and final states:
$$T_i\, V_i^{\gamma-1} = T_f\, V_f^{\gamma-1}\,.$$
Since the gas is compressed to one-fourth of its volume, we have
$$V_f = \frac{1}{4} V_i\,.$$
Substitute this into the adiabatic relation:
$$273\,V_i^{\gamma-1} = T_f \left(\frac{1}{4} V_iight)^{\gamma-1}\,.$$
With $$\gamma = \frac{3}{2},$$
$$\gamma - 1 = \frac{3}{2} - 1 = \frac{1}{2}\,.$$
So the relation becomes:
$$273\,V_i^{1/2} = T_f \left(\frac{1}{4}ight)^{1/2} V_i^{1/2}\,.$$
Cancel the common factor $$V_i^{1/2}$$ (provided it is nonzero) and simplify:
$$273 = T_f \left(\frac{1}{2}ight)\,.$$
Now solve for the final temperature $$T_f$$:
$$T_f = 273 	imes 2 = 546\,\mathrm{K}\,.$$
The change in temperature is then:
$$\Delta T = T_f - T_i = 546 - 273 = 273\,\mathrm{K}\,.$$
So, the change in temperature due to the thermodynamic process is $$\boxed{273\,\mathrm{K}}.$$

Thermodynamics & KTG

An ideal gas initially at 0^{\circ} \mathrm{C} temperature, is compressed suddenly to one fourth of its volume. If the ratio of specific heat at constant pressure to that at constant volume is 3 / 2, the change in temperature due to the thermodynamic process is _________ K.

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