The rms speed of the atoms in a 2.0 g sample of helium gas is 700 m/s. What is the thermal energy of the gas?

Verified step by step guidance

1. Step 1: Recall the formula for the root-mean-square (rms) speed of gas particles: $\sqrt{\frac{3kT}{m}}$, where $k$ is the Boltzmann constant, $T$ is the temperature, and $m$ is the mass of a single atom.
2. Step 2: Rearrange the formula to solve for the temperature $T$: $T=\frac{m{v}^2}{3k}$, where $v$ is the rms speed.
3. Step 3: Determine the mass of a single helium atom. The molar mass of helium is approximately 4.00 g/mol, and the mass of one atom is $\frac{4.00\times 10{-3}^3}{6.022\times 10{23}^{}}$ kg.
4. Step 4: Use the formula for the total thermal energy of a monatomic ideal gas: $E=\frac{3nRT}{2}$, where $n$ is the number of moles, $R$ is the gas constant, and $T$ is the temperature.
5. Step 5: Calculate the number of moles in the 2.0 g sample of helium using $n=\frac{m}{M}$, where $m$ is the mass of the sample and $M$ is the molar mass. Substitute the values into the thermal energy formula to find the result.

Key Concepts

Root Mean Square Speed

The root mean square (rms) speed is a statistical measure of the speed of particles in a gas. It is calculated as the square root of the average of the squares of the speeds of the individual particles. This concept is crucial for understanding the kinetic energy of gas particles, as it relates directly to their thermal motion and energy distribution.

Thermal Energy

Thermal energy refers to the total kinetic energy of the particles in a substance due to their motion. In the context of gases, it can be calculated using the formula E = (3/2)nRT, where n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. Understanding thermal energy is essential for analyzing the energy content of gases and their behavior under different conditions.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that relates the pressure, volume, temperature, and number of moles of an ideal gas. It is expressed as PV = nRT. This law is important for calculating various properties of gases, including thermal energy, and helps in understanding how changes in one variable affect the others in a gas system.