Question:
5 moles of unknown gas is heated at constant volume from 10°C to 20°C. The molar specific heat of this gas at constant pressure \( c_p = 8 \) cal/mol·°C and \( R = 8.36 \) J/mol·°C. The change in the internal energy of the gas is _______ calorie.}
5 moles of unknown gas is heated at constant volume from 10°C to 20°C. The molar specific heat of this gas at constant pressure \( c_p = 8 \) cal/mol·°C and \( R = 8.36 \) J/mol·°C. The change in the internal energy of the gas is _______ calorie.}
Updated On: Apr 10, 2026
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Correct Answer: 300
Solution and Explanation
Step 1: Understand the given quantities.
We are given:
- \( n = 5 \) moles (amount of the gas),
- \( c_p = 8 \) cal/mol·°C (molar specific heat at constant pressure),
- The temperature change \( \Delta T = 20°C - 10°C = 10°C \),
- \( R = 8.36 \) J/mol·°C (the universal gas constant).
We need to find the change in the internal energy of the gas.
Step 2: Use the first law of thermodynamics.
The first law of thermodynamics states that the change in internal energy \( \Delta U \) at constant volume is given by: \[ \Delta U = n \cdot c_V \cdot \Delta T \] where \( c_V \) is the molar specific heat at constant volume, and \( \Delta T \) is the temperature change.
Step 3: Relate \( c_V \) and \( c_p \).
We know that: \[ c_p - c_V = R \] Thus, we can find \( c_V \): \[ c_V = c_p - R = 8 \, \text{cal/mol·°C} - \frac{8.36 \, \text{J/mol·°C}}{4.18 \, \text{J/cal}} = 8 - 2 = 6 \, \text{cal/mol·°C}. \]
Step 4: Calculate the change in internal energy.
Now we can calculate the change in internal energy: \[ \Delta U = n \cdot c_V \cdot \Delta T = 5 \cdot 6 \cdot 10 = 300 \, \text{calories}. \]
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