One mole of perfect gas expands adiabatically. As a result of this it's pressure Temperature and volume change from p1 v1 T1 to p2 v2 T2 respectively. If molar specific heat at constant volume is Cv then the magnitude of work done by the gas is?

Question

Answer

Adiabatic Expansion of One Mole of Perfect Gas

Given Information:

One mole of perfect gas

Adiabatic expansion

Initial State: p1, v1, T1

Final State: p2, v2, T2

Molar specific heat at constant volume: Cv

Work Done by the Gas:

The magnitude of work done by the gas during adiabatic expansion can be calculated using the following formula:

W = (Cv * (T2 - T1)) - (R * (T2 * ln(v2/v1) - T1 * ln(p2/p1)))

Cv: Molar specific heat at constant volume

T1, T2: Initial and final temperatures

v1, v2: Initial and final volumes

p1, p2: Initial and final pressures

R: Universal gas constant

Explanation:

When a gas expands adiabatically, no heat is exchanged between the system and the surroundings. Therefore, the process is isentropic, i.e. the entropy of the system remains constant. The work done by the gas is equal to the decrease in its internal energy. The formula for work done takes into account the change in temperature and volume of the gas.

The first term (Cv * (T2 - T1)) represents the work done due to the change in temperature. The second term (R * (T2 * ln(v2/v1) - T1 * ln(p2/p1))) represents the work done due to the change in volume and pressure. The constant R is the universal gas constant.

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