Impact of Pressure Doubling on the Root Mean Square Velocity of a Gas at Constant Temperature

The relationship between the root mean square (RMS) velocity of a gas and its pressure is a fundamental concept in thermodynamics. This article delves into the changes in RMS velocity when the pressure of a gas sample is doubled while maintaining a constant temperature, exploring the underlying principles and providing a detailed analysis.

Key Relationships

To understand how the RMS velocity of a gas sample changes when pressure is doubled at constant temperature, we need to consider the key relationships, particularly the ideal gas law and the formula for RMS velocity.

The Ideal Gas Law

The ideal gas law is given by:

PV nRT

Where:

P is pressure V is volume n is the number of moles R is the ideal gas constant T is temperature

RMS Velocity

The root mean square (RMS) velocity of a gas is given by:

v_{rms} sqrt{frac{3RT}{M}}

Where:

M is the molar mass of the gas

Analysis

When the pressure of a gas is doubled at constant temperature, we can express this mathematically:

Initial pressure P Initial RMS velocity v_{rms} When the pressure is doubled, P 2P

Since the temperature remains constant, the RMS velocity depends on temperature (T) and molar mass (M) but not directly on pressure. Because the temperature is constant, the RMS velocity remains dependent on frac{RT}{M}.

Conclusion

Thus, when the pressure is doubled at constant temperature, the RMS velocity of the gas does not change. Therefore, the root mean square velocity remains the same:

v_{rms} v_{rms}

Discussion on Molecular Velocities

The velocity of a gaseous molecule, whether it is average, root mean square, or most probable, is directly proportional to the temperature and inversely proportional to the molecular mass of the gas. The RMS velocity of a gas is given by:

v_{rms} sqrt{frac{3RT}{M}}

At constant temperature, this formula implies that the RMS velocity is not affected by changes in pressure, as long as the temperature remains constant.

Relationship Between RMS Velocity and Pressure

It is often misconceived that pressure directly affects the velocity of gas molecules. In reality, when the pressure of a gas is increased, its volume decreases, maintaining the product PV nRT constant. This change in volume indirectly impacts the density (d) of the gas, which is represented by:

v_{rms} sqrt{frac{3PV}{M}} sqrt{frac{3P}{d}}

At a constant temperature, P/d remains a constant, further supporting the idea that the RMS velocity is not directly influenced by pressure.

Conclusion

In summary, the RMS velocity of a gas sample remains unchanged when the pressure is doubled at constant temperature. This occurs because the RMS velocity is a function of temperature and molar mass, not pressure. Understanding these principles helps in comprehending the behavior of gases under different conditions.