Question

If the internal energy of 3 moles of a gas at a temperature of 27 °C is 2250R, then the number of degrees of freedom of the gas is (R - Universal gas constant)

• A. \(3 \)
• B. \(5 \)
• C. \(4 \)
• D. \(6 \)

Hint:

The internal energy of an ideal gas is directly related to its degrees of freedom \(f\) and temperature \(T\). Formula: \(U = \frac{f}{2} nRT\), where \(n\) is the number of moles and \(R\) is the universal gas constant. Degrees of Freedom: Monatomic gas (e.g., He, Ne, Ar): \(f = 3\) (translational only) Diatomic gas (e.g., O\(_2\), N\(_2\), H\(_2\)): \(f = 5\) (3 translational + 2 rotational at ordinary temperatures) Polyatomic gas (non-linear, e.g., NH\(_3\), CH\(_4\)): \(f = 6\) (3 translational + 3 rotational) Vibrational degrees of freedom become active at higher temperatures. Always ensure temperature is in Kelvin when using gas laws.

Answer 1

The Correct Option is B

Step 1: Identify the given parameters and convert units if necessary.
Number of moles of gas, \(n = 3\) moles.
Temperature of the gas, \(T = 27^\circ \text{C}\).
Convert the temperature from Celsius to Kelvin:
\(T_K = T_{^\circ \text{C}} + 273.15 = 27 + 273 = 300 \, \text{K}\).
Internal energy of the gas, \(U = 2250R\), where \(R\) is the universal gas constant.
Step 2: Recall the formula for the internal energy of an ideal gas.
The internal energy \(U\) of an ideal gas with \(n\) moles and \(f\) degrees of freedom at temperature \(T\) is given by the formula: \[ U = \frac{f}{2} nRT \] where \(f\) is the number of degrees of freedom. 
Step 3: Substitute the given values into the formula and solve for \(f\).
We have \(U = 2250R\), \(n = 3\), and \(T = 300 \, \text{K}\). Substitute these values into the internal energy formula: \[ 2250R = \frac{f}{2} (3 \, \text{moles}) (R) (300 \, \text{K}) \] Notice that the universal gas constant \(R\) appears on both sides of the equation, so it can be cancelled out: \[ 2250 = \frac{f}{2} (3 \times 300) \] \[ 2250 = \frac{f}{2} (900) \] \[ 2250 = 450f \] Now, solve for \(f\): \[ f = \frac{2250}{450} \] \[ f = 5 \] 
Step 4: Conclude the number of degrees of freedom.
The number of degrees of freedom of the gas is 5. This value typically corresponds to a diatomic gas (e.g., O\(_2\), N\(_2\)) at moderate temperatures, where it has 3 translational degrees of freedom and 2 rotational degrees of freedom. The final answer is \( \boxed{5} \).