Question

One kilogram of a diatomic gas is at a pressure of \(2 \times 10^4 { N/m}^{-2}\) and the density of the gas is \(4 { kg/m}^{-3}\). The internal energy of this gas:

• A. \(1.25 \times 10^4 { J}\)
• B. \(2.4 \times 10^4 { J}\)
• C. \(2.8 \times 10^4 { J}\)
• D. \(1.25 \times 10^6 { J}\)

Hint:

Always check the consistency of units and consider common room temperature assumptions when specific conditions are not given.

Answer 1

The Correct Option is A

We are given:
Pressure \( P = 2 \times 10^4 \, {N/m}^2 \),
Density \( \rho = 4 \, {kg/m}^3 \),
Mass \( m = 1 \, {kg} \).
For a diatomic gas, the internal energy can be calculated using the formula for the internal energy of an ideal gas: \[ U = \frac{f}{2} nRT \] where \( f \) is the number of degrees of freedom (for a diatomic gas, \( f = 5 \)), \( n \) is the number of moles, and \( R \) is the universal gas constant (\( R = 8.314 \, {J/mol} \, {K} \)). However, we can also use the equation \( P = \frac{\rho R}{M} T \), where \( M \) is the molar mass of the gas. Rearranging, we get: \[ T = \frac{P M}{\rho R} \] Now, substituting the given values: - \( P = 2 \times 10^4 \, {N/m}^2 \), - \( \rho = 4 \, {kg/m}^3 \), - \( M = 4 \, {g/mol} = 0.004 \, {kg/mol} \). We calculate \( T \): \[ T = \frac{2 \times 10^4 \times 0.004}{4 \times 8.314} = \frac{80}{33.256} \approx 2.4 \, {K} \] Next, we calculate the internal energy using the formula for the internal energy of a diatomic gas: \[ U = \frac{f}{2} m R T \] Substituting the values for \( f = 5 \), \( m = 1 \, {kg} \), and \( T = 2.4 \): \[ U = \frac{5}{2} \times 1 \times 8.314 \times 2.4 \approx 1.25 \times 10^4 \, {J} \] Thus, the internal energy of the gas is \( 1.25 \times 10^4 \, {J} \), so the correct answer is (1). 
Conclusion: The internal energy of the gas is \( 1.25 \times 10^4 \, {J} \), so the correct answer is (1).