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Chapter 19: Problem 49

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 \(2.00 \mathrm{~K}\).

Short Answer

Expert verified
Answer: The change in internal energy is 41.57 J.

Step by step solution

01

Identify the Formula for Internal Energy

For an ideal gas, the change in internal energy (鈭哢) is given by the formula: 鈭哢 = n * c_v * 鈭員 where n is the number of moles, c_v is the specific heat capacity at constant volume, and 鈭員 is the change in temperature.
02

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).
03

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 鈭員 = 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. 鈭哢 = (1.00 \,\text{mol}) \times \left(\dfrac{5}{2} \times 8.314 \, \dfrac{\text{J}}{\text{mol K}}\right) \times (2.00 \, \text{K})
04

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: 鈭哢 = 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.

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