The rms speed of the molecules in 1.7 g of hydrogen gas is 1700 m/s.

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

What is the thermal energy of the gas?

500 J of work are done to compress the gas while, in the same process, 1700 J of heat energy are transferred from the gas to the environment. Afterward, what is the rms speed of the molecules?

Thank you!

300KJ

To answer the questions, let's break them down one by one.

Question 1: What is the total translational kinetic energy of the gas molecules?

To find the total translational kinetic energy, we need to use the formula:

KE = (1/2) * m * v^2

Where:
- KE is the kinetic energy
- m is the mass of the gas molecules
- v is the root mean square (rms) speed of the gas molecules

First, we need to find the number of moles of hydrogen gas using its molar mass. The molar mass of hydrogen (H2) is approximately 2 g/mol. Therefore, we can calculate the number of moles as follows:

Number of moles = mass / molar mass
Number of moles = 1.7 g / 2 g/mol
Number of moles = 0.85 mol

Next, we need to calculate the mass of one molecule of hydrogen gas (H2). Since we have 0.85 mol, we can use Avogadro's number to convert moles to molecules:

Number of molecules = number of moles * Avogadro's number
Number of molecules = 0.85 mol * 6.022 x 10^23 molecules/mol

Now, let's calculate the total translational kinetic energy:

KE = (1/2) * m * v^2
KE = (1/2) * (mass of one molecule) * (rms speed)^2
KE = (1/2) * (mass of hydrogen molecule) * (rms speed)^2
KE = (1/2) * (2 g/mol) * (rms speed)^2

Substituting the given rms speed:

KE = (1/2) * (2 g/mol) * (1700 m/s)^2

Now, we can calculate the answer to the first question.

Question 2: What is the thermal energy of the gas?

The thermal energy of the gas is equal to the total internal energy of the gas. It includes translational, rotational, and vibrational kinetic energies of the molecules. For an ideal gas, the internal energy is solely due to the translational kinetic energy of the gas molecules.

Therefore, the thermal energy is equal to the total translational kinetic energy calculated in question 1.

Question 3: Afterward, what is the rms speed of the molecules?

To calculate the final rms speed of the molecules after compression and heat transfer, we need to consider the work done on the gas and the heat transferred to the environment.

The change in internal energy (ΔU) of a system is given by the equation:

ΔU = Q - W
Where:
- ΔU is the change in internal energy
- Q is the heat added to the system
- W is the work done on the system

In this case, the work done on the system is positive since work is done to compress the gas. Therefore:

ΔU = Q - W
ΔU = 1700 J - 500 J
ΔU = 1200 J

Since the internal energy is solely due to translational kinetic energy, we can write:

ΔU = ΔKE = (1/2) * m * (v^2 - v₀^2)

Where:
- m is the mass of the gas molecules
- v is the final rms speed of the molecules
- v₀ is the initial rms speed of the molecules

We can rearrange the equation to solve for v:

v^2 = v₀^2 + (2 * ΔU) / m

Plugging in the given values:

v^2 = (1700 m/s)^2 + (2 * 1200 J) / m
v^2 = 2890000 m^2/s^2 + (2400 J) / m

Finally, we can calculate the answer to the third question by finding the square root of the equation above.

I hope this helps! Let me know if you have any further questions.