Question

Identify the correct equation relating \( \Delta H \), \( \Delta U \), and \( \Delta T \) for 1 mole of an ideal gas (R = gas constant):

• A. \( (\Delta H)^2 = \Delta U + R \Delta T \)
• B. \( \Delta H = (\Delta U)^2 + R \Delta T \)
• C. \( \Delta U = \Delta H - R \Delta T \)
• D. \( \Delta U = \Delta H + R \Delta T \)

Hint:

For an ideal gas, the relationship between enthalpy, internal energy, and temperature change follows \( \Delta H = \Delta U + R \Delta T \).

Answer 1

The Correct Option is C

Step 1: Apply the First Law of Thermodynamics for an Ideal Gas For an ideal gas: \[ \Delta H = \Delta U + R \Delta T \] Rearrange to solve for \( \Delta U \): \[ \Delta U = \Delta H - R \Delta T \] Thus, the correct answer is \( \Delta U = \Delta H - R \Delta T \).

Answer 2

For 1 mole of an ideal gas, the relationship between enthalpy change (\( \Delta H \)), internal energy change (\( \Delta U \)), and temperature change (\( \Delta T \)) can be derived from thermodynamic principles.

Step 1: Basic thermodynamic relation:
Enthalpy (\( H \)) is defined as:
\[ H = U + PV \] where \( U \) is internal energy, \( P \) is pressure, and \( V \) is volume.

Step 2: Change in enthalpy:
For changes,
\[ \Delta H = \Delta U + \Delta (PV) \] For 1 mole of ideal gas, using the ideal gas law:
\[ PV = RT \] where \( R \) is the gas constant and \( T \) is temperature.

Step 3: Substitute \( PV = RT \) into the expression:
\[ \Delta H = \Delta U + \Delta (RT) = \Delta U + R \Delta T \] Rearranging for \( \Delta U \),
\[ \Delta U = \Delta H - R \Delta T \]

Step 4: Interpretation:
- \( \Delta U \) represents the change in internal energy.
- \( \Delta H \) represents the change in enthalpy.
- The term \( R \Delta T \) accounts for the work done due to expansion or compression at constant pressure for 1 mole of ideal gas.

Therefore, the correct equation relating \( \Delta H \), \( \Delta U \), and \( \Delta T \) for 1 mole of an ideal gas is:
\[ \boxed{\Delta U = \Delta H - R \Delta T} \]