The work done by one mole of a monatomic ideal gas (gamma=5/3) in expanding adiabatically is 964 J. The initial temperature and volume of the gas are 395 K and .0848 m^3. Obtain a) the final temperature and b) the final volume of the gas.

To solve this problem, we can use the relation between work done and the change in temperature of an ideal gas in an adiabatic process. The relation is given by:

W = (Cv - Cp) * n * ΔT

where:
- W is the work done by the gas
- Cv is the molar heat capacity at constant volume
- Cp is the molar heat capacity at constant pressure
- n is the number of moles of gas
- ΔT is the change in temperature of the gas

In this case, we are given the work done W, the initial temperature Ti, and the initial volume Vi. We are required to find the final temperature Tf and the final volume Vf.

First, let's calculate the number of moles of gas using the ideal gas equation:

PV = nRT

where:
- P is the pressure (which is not given in the problem)
- V is the volume
- n is the number of moles
- R is the ideal gas constant
- T is the temperature

Assuming that the pressure remains constant throughout the process, we can rearrange the equation to solve for n:

n = PV / RT

Since the problem doesn't provide the pressure, we will use the given information to solve for n later.

Now, let's calculate the change in temperature ΔT:

ΔT = Tf - Ti

To find Tf, we need to rearrange the equation for work done W:

W = (Cv - Cp) * n * ΔT

Rearranging, we have:

ΔT = W / [(Cv - Cp) * n]

Now we have all the necessary information to solve for Tf. Plug in the values into the equation and calculate.

Next, to find the final volume Vf, we can use the adiabatic process equation:

(Ti / Tf) = (Vi / Vf)^(γ-1)

where γ is the ratio of specific heats for the monatomic gas (γ = 5/3).

Rearranging the equation to solve for Vf:

Vf = Vi * (Ti / Tf)^((γ-1)/γ)

Now, we can substitute the values of Ti, Tf, and Vi into the equation and calculate Vf.

To summarize:
a) Calculate Tf using the equation ΔT = W / [(Cv - Cp) * n]
b) Calculate Vf using the equation Vf = Vi * (Ti / Tf)^((γ-1)/γ)

Note: In this case, we don't have enough information to solve for the absolute values of Tf and Vf without knowing the value of pressure. However, we can calculate their ratio using the given values.

To solve this problem, we can use the equations for adiabatic processes in ideal gases:

1. For an adiabatic process, we have the relationship:
P1 * V1^gamma = P2 * V2^gamma

2. The work done by the gas is given by the equation:
W = (gamma / (gamma - 1)) * (P2 * V2 - P1 * V1)

Given information:
- gamma = 5/3
- W = 964 J
- T1 = 395 K
- V1 = 0.0848 m^3

Now, let's solve for the final temperature and final volume of the gas.

Step 1: Calculate the initial pressure (P1)
- We can use the ideal gas law: PV = nRT
- Since we have one mole of gas, n = 1 mole
- R is the ideal gas constant = 8.314 J/(mol K)
- Plugging in the values: P1 * V1 = n * R * T1
- P1 = (1 * 8.314 * 395) / 0.0848

Step 2: Calculate the final pressure (P2)
- Rearrange the adiabatic relationship equation: P2 = P1 * (V1 / V2)^(gamma)
- P2 = P1 * (V1 / V2)^(5/3)
- We can rearrange this equation to solve for V2: V2 = (V1 * (P1 / P2)^(3/5))

Step 3: Solve for the final temperature (T2)
- Use the ideal gas law with the final volume and pressure:
P2 * V2 = n * R * T2
- T2 = (P2 * V2) / (n * R)

Step 4: Calculate the final volume (V2)
- We already have the equation to solve for V2 from step 2.

Now, let's plug in the values and calculate:

Step 1: Calculate P1
- P1 = (1 * 8.314 * 395) / 0.0848

Step 2: Calculate P2 and V2
- P2 = P1 * (V1 / V2)^(5/3)
- V2 = V1 * (P1 / P2)^(3/5)

Step 3: Calculate T2
- T2 = (P2 * V2) / (n * R)

Step 4: Calculate V2
- V2 = V1 * (P1 / P2)^(3/5)

Now, let's calculate the values.