A 1.20 mol sample of an ideal diatomic gas at a pressure of 1.20 atm and temperature 380 K undergoes a process in which its pressure increases linearly with temperature. The final temperature and pressure are 680 K and 1.83 atm. (assume 5 active degrees of freedom)
b. Determine the work done by the gas.
(Since volume, pressure, and temperature all change, I'm not sure where to go from there with PV=nRT.)
To determine the work done by the gas in this case, we can use the equation for work done for an ideal gas undergoing a process where both pressure and volume change:
W = ∫PdV
In this case, we are given that the pressure of the gas increases linearly with temperature, which means that the relationship between pressure and temperature can be expressed as:
P = mT + C
where m is the slope of the linear relationship and C is the y-intercept.
To solve for m and C, we can use the given initial and final conditions. At the initial condition, where T = 380 K and P = 1.20 atm, we can substitute these values into the equation to get:
1.20 atm = m(380 K) + C
Similarly, at the final condition, where T = 680 K and P = 1.83 atm, we can substitute these values into the equation to get:
1.83 atm = m(680 K) + C
Now we have a system of two equations with two unknowns (m and C). We can solve this system of equations to find the values of m and C.
Once we have the values of m and C, we can substitute them back into the equation P = mT + C. Using the ideal gas law, PV = nRT, we can solve for V in terms of T and P. Then we can differentiate this equation with respect to T to find dV. Finally, we can substitute P and dV into the equation for work done (W = ∫PdV) and integrate it to find the work done by the gas.
Since this involves a bit of math and calculations, I suggest using a scientific calculator or a mathematical software tool like MATLAB or Python to solve the system of equations and perform the integration.