Ideal gas law and partial derivatives?
Hi everyone interesting homework question I know the general way of doing it however am not sure how to solve it. The ideal gas law PV=nRT where P(pressure) 10^5 pascals and increasing at a rate of 10^4 pascals per second and T (temp) is 300K and increasing at a rate of 20K per second. Also n=0.25 and R is 8.314. At what rate is the volume changing and is it increasing or decreasing? I know I need to use partial differentiation however am not sure how to apply it for this question. Any help would be great thanks
Update:Sorry is the answer i have correct or not? Thanks for all the help given so far
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No need to use partial derivatives, here's a simple solution:
Rearrange the equation to end up with V on one side:
V = nRT / P
You're looking for the rate at which V is changing i.e. Delta V (or V2 - V1).
For a 1-second interval: Assume an initial condition (1).
V1 = nR(T1) / (P1) where T1 = 300K and P1 = 10^5 Pa. Solve for V1.
For a 1-second interval (2)
V2 = nR(T2) / (P2) where T2 = 300 + 20 and P2 = 10^5 + 10^4. Solve for V2.
Rate of change is (V2 - V1) per second.
PV = nRT
P, T, n, R are given, find V
differentiate with respect to time t, note that n and R are fixed
P(dV/dt) + V(dP/dt) = nR (dT/dt)
P, dP/dt, dT/dt, n and R are all given, and you just found V, find dV/dt.
they are asking how one variable ameliorations with a version in yet another. on your occasion how T varies with changing x and t. you may rearrange PV = nRT to be a function in the two P, V or T then take the spinoff with understand to the V, T and P.