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. Identify the mass of the helium gas sample and convert it to kilograms. Since the mass given is 2.0 g, you need to convert this mass into kilograms by dividing by 1000.
2. Calculate the number of moles of helium in the sample. Use the molar mass of helium, which is approximately 4.00 g/mol. Divide the mass in kilograms by the molar mass to find the number of moles.
3. Use the formula for the root mean square (rms) speed of the gas particles, which is given by \( v_{rms} = \sqrt{\frac{3kT}{m}} \), where \( k \) is the Boltzmann constant, \( T \) is the temperature, and \( m \) is the mass of a single particle. Rearrange this formula to solve for the temperature \( T \).
4. Calculate the thermal energy of the gas using the formula \( E = \frac{3}{2} nRT \), where \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature calculated in the previous step.
5. Substitute the values of \( n \) and \( T \) into the thermal energy formula to find the thermal energy of the helium gas.

Key Concepts

Root Mean Square Speed (rms speed)

The root mean square speed is a measure of the average speed of particles in a gas, calculated as the square root of the average of the squares of the speeds of the individual particles. It is particularly useful in kinetic theory, as it relates to the temperature and energy of the gas. For an ideal gas, the rms speed can be expressed as v_rms = sqrt(3kT/m), where k is the Boltzmann constant, T is the temperature, and m is the mass of a gas particle.

Thermal Energy

Thermal energy is the total kinetic energy of the particles in a substance due to their motion. In the context of an ideal gas, it can be calculated using the formula E = (3/2)nRT, where E is the thermal energy, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. This energy is directly related to the temperature of the gas and reflects the energy available for doing work or transferring heat.

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in thermodynamics that describes the behavior of ideal gases. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. This law allows for the calculation of various properties of gases, including their thermal energy, by relating pressure, volume, and temperature in a single equation.