Question

The correct relation(s) for an ideal gas in a closed system is/are

• A. \((\frac{∂H}{∂V})_T=0\)
• B. \((\frac{∂T}{∂P})_H=0\)
• C. \((\frac{∂H}{∂P})_T=0\)
• D. \((\frac{∂H}{∂T})_P=0\)

Answer 1

The Correct Option is A, B, C

To solve this question, we need to understand the relationships between thermodynamic variables for an ideal gas in a closed system. The variables involved here are enthalpy (\(H\)), pressure (\(P\)), temperature (\(T\)), and volume (\(V\)). Let's analyze each option:

1. \((\frac{∂H}{∂V})_T=0\):
   For an ideal gas, enthalpy (\(H = U + PV\)) depends only on temperature (\(T\)) because the internal energy (\(U\)) and the product (\(PV\)) both depend only on \(T\) in an ideal gas. Therefore, at constant temperature, a change in volume does not affect the enthalpy of an ideal gas, making this statement true.
2. \((\frac{∂T}{∂P})_H=0\):
   At constant enthalpy for an ideal gas, there is no direct dependency between temperature and pressure. The ideal gas law (\(PV=nRT\)) implies that temperature changes would require changes in pressure and volume proportionally, but under constant \(H\), any pressure change does not directly impact temperature. Hence, this relation holds true.
3. \((\frac{∂H}{∂P})_T=0\):
   At constant temperature for an ideal gas, enthalpy is unaffected by changes in pressure since enthalpy is only a function of temperature in this case. Thus, this relation is true.
4. \((\frac{∂H}{∂T})_P=0\):
   This is not valid for an ideal gas, as enthalpy (\(H = U + PV\)) does change with temperature even at constant pressure, because internal energy and the product \(PV\) both change with \(T\).

Thus, the correct relations for an ideal gas in a closed system are: \((\frac{∂H}{∂V})_T=0\), \((\frac{∂T}{∂P})_H=0\), and \((\frac{∂H}{∂P})_T=0\).