Problem 49 - University Physics with Modern Physics

Question

Calculate the change in internal energy of 1.00 mole of a diatomic ideal gas that starts at room temperature \((293 \mathrm{~K})\) when its temperature is increased by \(2.00 \mathrm{~K}\).

Step-by-Step Solution

Answer

Answer: The change in internal energy is 41.57 J.

Step 1: Identify the Formula for Internal Energy

For an ideal gas, the change in internal energy (∆U) is given by the formula: ∆U = n * c_v * ∆T where n is the number of moles, c_v is the specific heat capacity at constant volume, and ∆T is the change in temperature.

Step 2: Determine the Specific Heat Capacity for a Diatomic Ideal Gas

For diatomic ideal gases, the specific heat capacity at constant volume (c_v) is given by: c_v = \dfrac{5}{2}R where R is the ideal gas constant and its value is 8.314 J/(mol K).

Step 3: Substitute the Given Values and Calculate the Change in Internal Energy

We are given the number of moles (n = 1.00 mol), the initial temperature T1 = 293 K, and the change in temperature ∆T = 2.00 K. We can now substitute these values along with the specific heat capacity at constant volume (c_v) into the formula for the change in internal energy. ∆U = (1.00 \,\text{mol}) \times \left(\dfrac{5}{2} \times 8.314 \, \dfrac{\text{J}}{\text{mol K}}\right) \times (2.00 \, \text{K})

Step 4: Evaluate the Expression and Find the Change in Internal Energy

Now we just need to perform the calculation to find the change in internal energy of the gas: ∆U = 1.00 \times \dfrac{5}{2} \times 8.314 \times 2.00 = 41.57 \, \text{J} Therefore, the change in internal energy of the 1.00 mole of diatomic ideal gas is 41.57 J when its temperature is increased by 2.00 K.