Question

Two litres of hydrogen at a pressure of 10^5 Pa expands adiabatically to 1.5 times the initial volume. Find the work done?

Answer 1

Answer on Question #79364 - Physics - Molecular Physics / Thermodynamics

Question: Two litres of hydrogen at a pressure of $10^{\wedge}5$ Pa expands adiabatically to 1.5 times the initial volume. Find the work done?

Answer:

The first law of thermodynamics [1] can be formulated in the following way:

$$Q = \Delta U + A,$$

where $Q$ is the amount of heat supplied to the system, $\Delta U$ is the change in the internal energy of the system, $A$ is the work done by the system. For adiabatic process $Q = 0$, so that the work can be done due to the loss of the internal energy only:

$$A = - \Delta U.$$

Hydrogen is a diatomic (ideal) gas, and each of its molecules possesses $i = 5$ degrees of freedom. Hence, the change in its internal energy should be written as:

$$\Delta U = \frac {5}{2} \nu R \left(T _ {2} - T _ {1}\right).$$

According to Mendeleev-Clapeyron equation

$$p _ {1} V _ {1} = \nu R T _ {1},$$ $$p _ {2} V _ {2} = \nu R T _ {2},$$

Hence,

$$\nu R \left(T _ {2} - T _ {1}\right) = p _ {2} V _ {2} - p _ {1} V _ {1}.$$

There is also one more useful relation for the adiabatic processes:

$$p V ^ {\gamma} = \text{Const},$$

where $\gamma$ is the adiabatic index (= heat capacity ratio). For diatomic ideal gases $\gamma = \frac{7}{5}$ (see [2] for details).

As a result, one can show that:

$$p _ {2} = p _ {1} \left(\frac {V _ {1}}{V _ {2}}\right) ^ {\gamma}.$$

Combining (2)-(3), (5), (7) together and taking into account the relation $V_{2} = 1.5V_{1} = \frac{3}{2} V_{1}$, we deduce

$$A = - \frac {5}{2} \left(p _ {2} V _ {2} - p _ {1} V _ {1}\right) = \frac {5}{2} p _ {1} V _ {1} \left[ 1 - \left(\frac {V _ {1}}{V _ {2}}\right) ^ {y - 1} \right] = \frac {5}{2} p _ {1} V _ {1} \left[ 1 - \left(\frac {2}{3}\right) ^ {2 / 5} \right].$$

Substituting the numerical values, we obtain:

$$A \approx \frac {5}{2} \cdot 1 0 ^ {5} \cdot 2 \cdot 1 0 ^ {- 3} (1 - 0. 8 5) = 4 2 5 J.$$

[1] (Electronic resource) https://en.wikipedia.org/wiki/First_law_of_thermodynamics

[2] (Electronic resource) https://en.wikipedia.org/wiki/Heat_capacity_ratio

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