Need help finding spring constant from volume, area, temperature and distance

[defmar]

An ideal gas is 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 m^2. The initial pressure, volume, and temperature of the gas are, respectively, P0, V0 = 6.00*10^-4 m^3 and T0 = 273 K, and the spring is initially stretched by an amount x0 = 0.800 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 = 0.1000 m with respect to its unstrained length. What is the final temperature of the gas?

For the life of me, I cannot find the spring constant k that is needed to solve the rest of the entire problem. I know that KE = (3/2)kT, but I don't know KE either. I'm not given the initial pressure, just the variable. Anyone able to help me find k? I'm lost on how to come up with it.

 

[Andrew Mason]

An ideal gas is 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 m^2. The initial pressure, volume, and temperature of the gas are, respectively, P0, V0 = 6.00*10^-4 m^3 and T0 = 273 K, and the spring is initially stretched by an amount x0 = 0.800 m with respect to its unstrained length.

You can't solve for k or n. But you do not have to find k or n.

Just write the expression for P0 in terms of x0, A, V0, n and T0 and similarly write the expression for Pf . When you solve for Tf you will see that k and n drop out.

AM

 

[defmar]

You can't solve for k or n. But you do not have to find k or n.

Just write the expression for P0 in terms of x0, A, V0, n and T0 and similarly write the expression for Pf . When you solve for Tf you will see that k and n drop out.

AM

Thank you.

I'm getting T_f = [T_0*(V_0+A*ΔX)*X_f/A] / [(X_0*V_0)/A] solving :)

 

[Andrew Mason]

Thank you.

I'm getting T_f = [T_0*(V_0+A*ΔX)*X_f/A] / [(X_0*V_0)/A] solving :)

What is Δx?

What is your expression for P in terms of k, x and A? What is Pf/P0?

What is Pf/P0 in terms of Tf, T0, Vf and V0?

Work out Tf from that.

AM

 

[asteriatic]

The spring constant, k, can be calculated using the formula k = (P0A)/(V0x0). This formula is derived from the ideal gas law, which states that PV = nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is temperature. In this case, the gas is an ideal gas, so we can assume that n and R are constant throughout the process. Therefore, we can rearrange the ideal gas law to get P = (nR/V)T.

Since the gas is confined to a cylinder with a massless piston, the pressure is directly proportional to the force exerted by the gas on the piston, which is balanced by the force exerted by the spring. This means that P = kx, where k is the spring constant and x is the displacement of the spring.

Using this information, we can substitute P = kx into the ideal gas law equation to get (kx)A = (nR/V)T. Rearranging this equation, we get k = (P0A)/(V0x0).

Now, we have all the necessary values to calculate the spring constant, k. Plugging in the given values, we get k = (P0A)/(V0x0) = (PfA)/(Vfxf).

From here, we can use the given value of xf = 0.1000 m to solve for Pf, which is the final pressure of the gas. Then, we can use the ideal gas law to solve for Tf, the final temperature of the gas.

In summary, the spring constant, k, can be calculated using the formula k = (P0A)/(V0x0). From there, we can use the given values to solve for the final pressure, Pf, and then use the ideal gas law to solve for the final temperature, Tf. I hope this helps with your problem!