The drawing shows an ideal gas confined to a cylinder by a massless piston that is attached to an ideal spring. Outside the cylinder is a vacuum. The cross-sectional area of the piston is A = 2.50 10-3 m2. The initial pressure, volume, and temperature of the gas are, respectively, P0, V0 = 6.00 10-4 m3 and T0 = 273 K, and the spring is initially stretched by an amount x0 = 0.0808 m with respect to its unstrained length. The gas is heated, so that its final pressure, volume, and temperature are Pf, Vf and Tf and the spring is stretched by an amount xf = 0.0930 m with respect to its unstrained length. What is the final temperature of the gas?

Fx Applied = k*x

(k=5.8x10^4)

Vf=Vo+A(Xf-Xo)

Fo=Po*A and Ff=Pf*A

Po=kXo/A and Pf=kXf/A

Tf=(Pf*Vf*To)/(Po*Vo)

Solve for Tf by plugging in the unknown values with the equations above.

It would look something like this...
Tf=(kxf/a)*(Vo+A(delta x))*(To)/(kxo/A)(Vo)

To determine the final temperature of the gas, we can use the ideal gas law equation combined with the changes in volume and pressure.

The ideal gas law equation is given by: PV = nRT

Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature

Since the gas is an ideal gas, the number of moles remains constant. Therefore, we can write the initial and final states of the gas as:

P0V0 = nRT0 ---(1)
PfVf = nRTf ---(2)

To find the final temperature (Tf), we need to eliminate the number of moles (n) from equations (1) and (2). We can do this by dividing equation (2) by equation (1):

(PfVf) / (P0V0) = (nRTf) / (nRT0)
PfVf / P0V0 = Tf / T0

Now, we can rearrange the equation to solve for final temperature (Tf):

Tf = (PfVf / P0V0) * T0

Given:
P0 = initial pressure = known value
V0 = initial volume = known value
T0 = initial temperature = known value
Pf = final pressure = unknown
Vf = final volume = unknown
Tf = final temperature = to be determined

We are given the values of all variables except Pf and Vf. However, we can relate the changes in volume and pressure to the displacement of the spring using Hooke's Law:

F = kx

Where:
F = force exerted by the spring
k = spring constant
x = displacement of the spring

From the problem statement, we know that the spring is initially stretched by an amount x0 and is stretched further by an amount xf. Therefore, the change in displacement (Δx) is given by:

Δx = xf - x0

The force exerted by the spring can be related to the pressure using the cross-sectional area of the piston:

F = P0A

Combining these equations, we get:

P0A = k(Δx)

Rearranging for k:

k = (P0A) / (Δx)

Now, we can substitute this value of k into the equation for force:

F = kx
P0A = (P0A / Δx) * x

Simplifying further, we get:

Px = P0Δx / x

Now, we can substitute this value of Px into the equation for final pressure (Pf):

PfVf = PxV0

Substituting Px from the previous equation:

PfVf = (P0Δx / x) * V0

Rearranging for Vf:

Vf = (P0Δx / x) * (V0 / Pf)

Now, we have expressed the final volume of the gas (Vf) in terms of the known values and the unknown final pressure (Pf).

Substituting this expression for Vf back into the equation for final temperature (Tf):

Tf = (Pf * (P0Δx / x) * (V0 / Pf)) / P0V0

Simplifying further:

Tf = (P0Δx * (V0 / x)) / P0V0

At this point, we can see that several terms (P0 and V0) cancel out, and we are left with:

Tf = Δx * (V0 / x)

Now, we can substitute the known values for x, x0, and V0, and calculate the final temperature (Tf).