Ideal gas law derivative problem

[dear2009]

Dear MathHelp Participants,


The ideal gas law states that

PV = nRT,

where P is pressure (in atmospheres), V is volume (in liters), n is the number of moles of gas atoms (a mole contains 6.02 x 1023 atoms), R is the gas constant (0.08206 L atm mol­–1 ­K­–1). Suppose 25 moles of helium is contained in a cylinder whose volume is decreasing at the rate of 1 L/sec, while the pressure is increasing at the rate of ½ atmosphere per second. How fast is the temperature changing when the pressure is 10 atm, the volume is 60 liters, and the temperature is 400K? (answers given are approximate, in units of degrees Kelvin per second)

I began with
V dP/dt + P(dV/dt) = RT dn/dt + nR (dT/dt)
When i setup this equation/derivative however, the numbers didnt add up correctly

60 (1/2) + 10(1) = (0.8206)(400) + 25 (0.8206) dT/dt
which didnt lead to my answer


any suggestions would be greatly appreciated

 

[wjm11]

PV = nRT,

where P is pressure (in atmospheres), V is volume (in liters), n is the number of moles of gas atoms (a mole contains 6.02 x 1023 atoms), R is the gas constant (0.08206 L atm mol–1 K–1). Suppose 25 moles of helium is contained in a cylinder whose volume is decreasing at the rate of 1 L/sec, while the pressure is increasing at the rate of ½ atmosphere per second. How fast is the temperature changing when the pressure is 10 atm, the volume is 60 liters, and the temperature is 400K? (answers given are approximate, in units of degrees Kelvin per second)

I began with
V dP/dt + P(dV/dt) = RT dn/dt + nR (dT/dt)
When i setup this equation/derivative however, the numbers didnt add up correctly

60 (1/2) + 10(1) = (0.8206)(400) + 25 (0.8206) dT/dt

Get organized first; right down things you know and rearrange your equation to solve for the variable in question. Remember that the derivative of a constant is 0, so the derivative of n is 0.

T = PV/nR
n = 25 moles (a constant)
R = .08206 (L)(atm)/(mol)(K) (a constant)
dV/dT = -1L/s
dP/dt = .5 atm/s
P = 10 atm
V = 60 L
T = 400 K

Find the derivative with respect to time:

T = PV/nR
DT/dt = [P(dV/dt) + V(dP/dt)]/nR

Just plug in the values.