Kinetic Theory of Gas
The Kinetic Theory of gases is a classical model of gases, according to which gases are composed of molecules/particles that are in random motion. While undergoing this random motion, kinetic energy in molecules can assume random velocity across all directions. It also says that the constituent particles/molecules undergo elastic collision, which means that the total kinetic energy remains constant before and after the collision. The average kinetic energy of the particles also determines the pressure of the gas.
P-V Diagram
A P-V diagram is a very important tool of the branch of physics known as thermodynamics, which is used to analyze the working and hence the efficiency of thermodynamic engines. As the name suggests, it is used to measure the changes in pressure (P) and volume (V) corresponding to the thermodynamic system under study. The P-V diagram is used as an indicator diagram to control the given thermodynamic system.
![**Problem Statement:**
4. **(13.49)** If the pressure in a gas is tripled while its volume is held constant, by what factor does \(v_{\text{rms}}\) change?
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### Explanation
Consider the given problem as part of a larger discussion on the relationships between pressure, volume, temperature, and the root mean square speed (\(v_{\text{rms}}\)) of gas molecules. According to the ideal gas law, the relationship between pressure (P), volume (V), and temperature (T) is given by:
\[ PV = nRT \]
where \( n \) is the number of moles of the gas, and \( R \) is the universal gas constant.
The root mean square speed (\(v_{\text{rms}}\)) of the gas molecules is related to the temperature by the equation:
\[ v_{\text{rms}} = \sqrt{\frac{3kT}{m}} \]
where \( k \) is the Boltzmann constant and \( m \) is the mass of a gas molecule.
Given that the volume is constant and the pressure is tripled, we can use the ideal gas law to determine how the temperature changes:
Since \( V \) is constant, from \( PV = nRT \):
\[ P_1 V = nR T_1 \]
After tripling the pressure:
\[ P_2 = 3P_1 \]
\[ P_2 V = nR T_2 \]
Substituting \( P_2 \) as \( 3P_1 \):
\[ 3P_1 V = nR T_2 \]
Since \( P_1 V = nR T_1 \), we can write:
\[ T_2 = 3T_1 \]
Thus, the temperature is tripled. Now, considering the \(v_{\text{rms}}\) equation:
\[ v_{\text{rms}} \propto \sqrt{T} \]
When the temperature is tripled:
\[ v_{\text{rms}_2} = \sqrt{3T_1} = \sqrt{3} \times \sqrt{T_1} \]
Therefore, when the pressure in the gas is tripled while its volume is held constant, the root mean square speed (\(v_{\text{rms}}\)) changes by a factor of \( \sqrt{3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F529e4fe6-e8c4-4722-88aa-75bc8f350b68%2F96991f18-6f23-4e1b-8dbc-40ec07fee42b%2Frx99rtj_processed.jpeg&w=3840&q=75)
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