Question

The rms speed of the molecules in 1.0 g of hydrogen gas is 1800 m/s.

a. What is the total translational kinetic energy of the gas molecules?

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b. What is the thermal energy of the gas?

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c. 500 J of work are done to compress the gas while, in the same process, 1200 J of heat energy are transferred from the gas to the environment. Afterward, what is the rms speed of the molecules?

Answer 1

The kinetic energy of the molecules in terms of root mean square velocity is given below:

Here, is the kinetic energy, is the mass of the molecule and is the root mean square velocity of the molecule.

The root mean square velocity is given below:

Here, is the gas constant, is the temperature, and is the molar mass.

The number of moles is given below:

Here, is the number of moles, is the given mass and is the molar mass.

The thermal energy of the molecules is given below:

Here, is the thermal energy of the molecules, is the gas constant and is the temperature.

(a)

The translational kinetic energy of the gas molecules is given by the expression below:

Substituting for and for .

Converting the value into .

Therefore, the required translational kinetic energy is .

(b)

The equation for kinetic energy is given below:

Substituting the value of as .

Rearranging the equation for number of moles.

Substituting the value of in the equation of .

The equation for thermal energy is given below:

Multiplying and dividing the equation by .

Substituting for .

Therefore, the required thermal energy of the gas is.

(c)

The first law of thermodynamics is given below:

Substituting for and for .

The energy transformation equation is given below:

Substituting for and for .

The equation of in terms of kinetic energy is given below:

…… (1)

The translational kinetic energy of the gas molecules is given by the expression below:

…… (2)

Substituting equation (2) in (1).

Rearrange the above equation for .

Substituting for and for .

Therefore, the required rms speed of the molecules is.