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.}

Answer 1

Correct Answer: 300

Step 1: Understanding the Concept:
The change in internal energy of an ideal gas depends only on its temperature change and molar specific heat at constant volume (\(c_v\)), regardless of the process path.
We must ensure all values are in consistent units (calories) before applying the formula.
Step 2: Key Formula or Approach:
Mayer's relation: \(c_p - c_v = R\).
Change in internal energy: \(\Delta U = n c_v \Delta T\).
Mechanical equivalent of heat: \(1 \text{ calorie} = 4.18 \text{ Joules}\).
Step 3: Detailed Explanation:
First, convert the universal gas constant \(R\) from Joules to calories.
\[ R = 8.36 \text{ J/mol.}^\circ\text{C} = \frac{8.36}{4.18} \text{ cal/mol.}^\circ\text{C} = 2 \text{ cal/mol.}^\circ\text{C} \] Next, calculate the molar specific heat at constant volume \(c_v\) using Mayer's relation.
\[ c_v = c_p - R = 8 \text{ cal/mol.}^\circ\text{C} - 2 \text{ cal/mol.}^\circ\text{C} = 6 \text{ cal/mol.}^\circ\text{C} \] Now, identify the given parameters for the process:
Number of moles \(n = 5\).
Temperature change \(\Delta T = 20^\circ\text{C} - 10^\circ\text{C} = 10^\circ\text{C}\).
Calculate the change in internal energy:
\[ \Delta U = n c_v \Delta T \] \[ \Delta U = 5 \times 6 \times 10 \] \[ \Delta U = 300 \text{ calories} \] Step 4: Final Answer:
The change in the internal energy is \(300 \text{ calorie}\).