Helium gas is in a cylinder that has rigid walls. If the...

Helium gas is in a cylinder that has rigid walls. If the pressure of the gas is $2.00 \mathrm{~atm},$ then the root-mean-square speed of the helium atoms is $v_{\mathrm{rms}}=176 \mathrm{~m} / \mathrm{s}$. By how much (in atmospheres) must the pressure be increased to increase the $v_{\mathrm{rms}}$ of the He atoms by $100 \mathrm{~m} / \mathrm{s}$ ? Ignore any change in the volume of the cylinder.

Key Concepts

Pressure-Temperature Relationship in Rigid Containers

In a rigid container, where volume remains constant, the ideal gas law indicates that the pressure of a gas is directly proportional to its temperature. An increase in temperature leads to a proportional increase in pressure. This relationship is critical when determining how adjustments in the pressure will affect the temperature and consequently the kinetic properties, such as the rms speed, of the gas particles.

Root-Mean-Square Speed

The root-mean-square (rms) speed is a measure of the average speed of particles in a gas, and it is given by the formula v?rms? = ?(3RT/M), where R is the gas constant, T is the temperature, and M is the molar mass of the gas. This concept links the temperature of a gas directly to the kinetic energy of its particles, showing that an increase in temperature (and thus pressure in a rigid container) will increase the rms speed of the gas particles.

Ideal Gas Law

The ideal gas law, stated as PV = nRT, relates the pressure, volume, and temperature of an ideal gas. In a rigid container where the volume is constant, any change in temperature directly results in a change in pressure. This concept is fundamental in understanding how thermal energy affects the pressure of a gas when external constraints like volume are fixed.

Kinetic Molecular Theory

The kinetic molecular theory explains the behavior of gas particles in terms of their motion. It establishes the connection between the microscopic motion of atoms or molecules and macroscopic properties such as pressure and temperature. This theory provides the foundation for understanding the relationship between the average kinetic energy of gas particles and the temperature of the gas.

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