An ideal monoatomic gas is confined by a spring loaded massless piston of cross section 8*10^-3 m². Initially the gas is at 300 K and occupies a volume of 2.4*10^-3 m³ and the spring is in its relaxed state. The gas is heated by an electric heater until the piston moves out slowly without friction by 0.1 m. Calculate:

(a) The final temperature of the gas and

(b) the heat supplied by the heater.

The force constant of the spring is 800 N/m, atm pressure is 1.0 * 10⁵ N/m². The cylinder and the piston are thermally insulated.

To tackle this problem, we need to apply the principles of thermodynamics, particularly focusing on the behavior of an ideal gas and the work done by the gas as it expands against a spring. Let's break this down step by step to find both the final temperature of the gas and the heat supplied by the heater.

Understanding the Initial Conditions

We start with an ideal monoatomic gas at an initial temperature (T1) of 300 K and an initial volume (V1) of 2.4 x 10^-3 m³. The cross-sectional area (A) of the piston is 8 x 10^-3 m², and the spring is initially relaxed. The force constant of the spring (k) is 800 N/m. The atmospheric pressure (P_atm) is 1.0 x 10⁵ N/m².

Calculating the Initial Pressure

First, we can find the initial pressure (P1) of the gas using the ideal gas law, which states:

P1V1 = nRT1

However, we need to determine the number of moles (n) of the gas. We can rearrange the equation to find P1:

P1 = nRT1 / V1

But we can also find P1 directly by considering the forces acting on the piston. The pressure exerted by the gas must balance the atmospheric pressure plus the pressure from the spring when the piston is at rest:

P1 = P_atm + (F_spring / A)

Where F_spring = k * x, and x is the displacement of the spring. Since the spring is relaxed initially, we can assume x = 0, so:

P1 = P_atm = 1.0 x 10⁵ N/m².

Final Conditions After Heating

When the gas is heated, the piston moves out by 0.1 m. The new volume (V2) can be calculated as:

V2 = V1 + A * displacement = 2.4 x 10^-3 m³ + (8 x 10^-3 m² * 0.1 m) = 2.4 x 10^-3 m³ + 8 x 10^-4 m³ = 3.2 x 10^-3 m³.

Finding the Final Temperature

Now, we can find the final temperature (T2) of the gas using the ideal gas law again:

P2V2 = nRT2

We need to find the final pressure (P2). The pressure exerted by the gas when the piston is at the new position is:

P2 = P_atm + (F_spring / A)

To find F_spring, we calculate the force exerted by the spring when the piston is displaced:

F_spring = k * x = 800 N/m * 0.1 m = 80 N.

Thus, the new pressure becomes:

P2 = 1.0 x 10⁵ N/m² + (80 N / 8 x 10^-3 m²) = 1.0 x 10⁵ N/m² + 1.0 x 10⁴ N/m² = 1.1 x 10⁵ N/m².

Now we can substitute P2 and V2 into the ideal gas law to find T2:

1.1 x 10⁵ N/m² * 3.2 x 10^-3 m³ = nR * T2.

We can find n using the initial conditions:

n = P1V1 / (RT1) = (1.0 x 10⁵ N/m² * 2.4 x 10^-3 m³) / (R * 300 K).

Assuming R = 8.314 J/(mol·K), we can calculate n:

n = (1.0 x 10⁵ * 2.4 x 10^-3) / (8.314 * 300) = 0.096 mol.

Now substituting n back into the equation for T2:

1.1 x 10⁵ * 3.2 x 10^-3 = 0.096 * 8.314 * T2.

Solving for T2 gives:

T2 = (1.1 x 10⁵ * 3.2 x 10^-3) / (0.096 * 8.314) = 400 K.

Calculating the Heat Supplied by the Heater

To find the heat supplied (Q), we can use the first law of thermodynamics, which states:

Q = ΔU + W

Where ΔU is the change in internal energy and W is the work done by the gas. For a monoatomic ideal gas, the change in internal energy is given by:

ΔU = (3/2)nR(T2 - T1).

Substituting the values we have:

ΔU = (3/2) * 0.096 mol * 8.314 J/(mol·K) * (400 K - 300 K) = 0.144 J.

Next, we calculate the work done by the gas as it expands against the spring