Rate of Change for gas pressure

Question

I've watched several videos and looked at similar problems and plugged in my numbers, but I still get the wrong answer. I'm not sure where I am going wrong.

The gas law for an ideal gas at absolute temperature T (in kelvins), pressure P (in atmospheres), and volume V (in liters) is PV = nRT,

where n is the number of moles of the gas and R = 0.0821

is the gas constant. Suppose that, at a certain instant, P = 7.0 atm

and is increasing at a rate of 0.15 atm/min and V = 12 L

and is decreasing at a rate of 0.17 L/min. Find the rate of change of T with respect to time (in K/min) at that instant if n = 10 mol.

(Round your answer to four decimal places.)

Answer 1

Hi Alexis,

Let's start by defining terms:

PV = nRT

P = Pressure (atm)

V = Volume (L)

n = Number of Moles (#)

R = Gas Constant (atm*L/(mol*K))

T = Temperature (K)

The problem gives several of these values:

P = 7

V = 12

n = 10

R = 0.0821

They don't give you T, but you can determine that from the other four values.

T = PV/nR

T = (7*12)/(10*0.0821)

T = 102.3143 K

We'll find out later we don't actually need it though.

They give you two other pieces of information:

"Pressure is increasing at a rate of 0.15 atm/min." This means dP/dt = 0.15

"Volume is decreasing at a rate of 0.17 L/min." This means dV/dt = -0.17.

It asks you to find "The rate of change of T with respect to time". So they're looking for dT/dt.

In order to generate these terms, we'll need to implicitly differentiate the ideal gas equation. We need to consider P, L, and T as variables. R and n can be considered constants.

PV = nRT

d/dt(PV = nRT)

We'll differentiate the sides one by one.

Since P and V are both variables, we need to apply the product rule to the left side.

d/dt(PV) = (1)*dP/dt*V + P*(1)*dV/dt

d/dt(PV) = V*dP/dt + P*dV/dt

On the right side of the equation, T is the only variable. You can consider "nR" as a single constant who's just along for the ride.

d/dt(nRT) = nr*dT/dt

So:

V*dP/dt + P*dV/dt = nr*dT/dt

Plug in all your known values.

P = 7

V = 12

n = 10

R = 0.0821

dP/dt = 0.15

dV/dt = -0.17

(12)*(0.15) + (7)*(-0.17) = (10)*(0.0821)*dT/dt

1.8 - 1.19 = 0.821*dT/dt

0.61 = 0.821*dT/dt

dT/dt = 0.7430 K/min

Is this the answer you got?