Open Question
A gas of 1.0 x 10²⁰ atoms or molecules has 1.0 J of thermal energy. Its molar specific heat at constant pressure is 20.8 J/ mol K. What is the temperature of the gas?
21. Kinetic Theory of Ideal Gases
Internal Energy of Gases
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A container filled with 2 mol of an ideal, monoatomic gas is has a total internal energy equal to the kinetic energy of a 0.008kg bullet travelling at 700 m/s. What is the temperature of the gas in Kelvin?
A
123.2 K
B
78.6 K
C
235.7 K
D
11.2 K
Verified step by step guidance1
Start by calculating the kinetic energy of the bullet using the formula: \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of the bullet (0.008 kg) and \( v \) is the velocity (700 m/s).
The internal energy \( U \) of an ideal monoatomic gas is given by \( U = \frac{3}{2}nRT \), where \( n \) is the number of moles (2 mol), \( R \) is the ideal gas constant (8.314 J/mol·K), and \( T \) is the temperature in Kelvin.
Set the kinetic energy of the bullet equal to the internal energy of the gas: \( \frac{1}{2}mv^2 = \frac{3}{2}nRT \).
Solve for the temperature \( T \) by rearranging the equation: \( T = \frac{mv^2}{3nR} \).
Substitute the known values into the equation to find the temperature: \( T = \frac{(0.008 \text{ kg})(700 \text{ m/s})^2}{3 \times 2 \text{ mol} \times 8.314 \text{ J/mol·K}} \).
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