Three moles of helium gas (molar mass M=4.00 g / mol ) are in a rigid container that keeps the

step 1 So this is a very simple problem. Here we will use some basic concepts. The first one is we have

Three moles of helium gas (molar mass M=4.00 g / mol ) are...

Three moles of helium gas (molar mass $M=4.00 \mathrm{~g} / \mathrm{mol}$ ) are in a rigid container that keeps the volume of the gas constant. Initially the rms speed of the gas atoms is $900 \mathrm{~m} / \mathrm{s}$. What is the rms speed of the gas atoms after $2400 \mathrm{~J}$ of heat energy is added to the gas?

Key Concepts

Heat Addition and Temperature Change

When heat energy is added to a gas in a rigid container (constant volume), the energy goes entirely into increasing the internal energy of the gas, as indicated by ?U = Q. This increase in internal energy raises the temperature of the gas. As temperature is directly linked to the rms speed through kinetic theory, any increase in temperature results in an increase in the rms speed, allowing one to relate the heat added to the change in molecular speeds.

Root Mean Square (rms) Speed

This concept refers to the measure of the average speed of gas molecules, calculated by taking the square root of the average of the squared speeds. In kinetic theory, the rms speed is directly related to the temperature of the gas through the equation v_rms = ?(3RT/M), where R is the ideal gas constant, T is the absolute temperature, and M is the molar mass. This relationship indicates that as the temperature increases, the rms speed of the gas particles also increases.

Internal Energy of an Ideal Gas

For a monatomic ideal gas, the internal energy is completely a function of temperature and can be expressed as U = (3/2)nRT, where n is the number of moles. This implies that any change in the internal energy (due to heat transfer in a constant volume process, for example) results in a change in temperature, and thereby alters the kinetic energy of the gas molecules and their rms speed.

Constant Volume Process

A constant volume process means that the volume of the gas does not change, so no work is done by the system. In such a scenario, all the heat energy added to the gas is used to increase its internal energy. This straightforward energy transfer allows the calculation of temperature change (and, consequently, the change in rms speed) directly from the amount of heat added.

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