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 = 0.091 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 = 0.12 m with respect to its unstrained length. What is the final temperature of the gas?
To determine the final temperature of the gas, we need to use the ideal gas law and the concept of work done.
1. First, let's determine the change in volume of the gas. The change in volume can be obtained by subtracting the initial volume from the final volume:
ΔV = Vf - V0
2. Next, we need to find the work done on the gas. The work done can be calculated using the formula:
Work = Force × Distance
In this case, the force exerted on the gas is due to the spring, and it is given by Hooke's Law:
Force = k × displacement
where k is the spring constant and displacement is the change in length of the spring.
3. The displacement of the spring can be calculated as the difference between the final and initial displacements:
d = xf - x0
4. The work done on the gas can be expressed as:
Work = -k × d
Note that the work done on the gas is negative because the gas is doing work on the surroundings.
5. The work done on the gas can also be expressed in terms of pressure and volume using the formula:
Work = P × ΔV
6. Equating the two expressions for work, we have:
-P × ΔV = -k × d
7. Rearranging the equation, we get:
P × ΔV = k × d
8. We can substitute the values given in the question to solve for the final pressure:
Pf × (Vf - V0) = k × (xf - x0)
9. Finally, we can rearrange the equation to solve for the final temperature:
Pf × Vf = (P0 × V0 × Tf) / T0
Tf = (Pf × Vf × T0) / (P0 × V0)
By substituting the given values for Pf, Vf, P0, V0, and T0, we can calculate the final temperature, Tf.