   Chapter 2.7, Problem 33E

Chapter
Section
Textbook Problem

# The gas law for an ideal gas at absolute temperature T (in kelvins), pressure P (in atmospheres, atm), and volume V (in liters) is P V   =   n R T , 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 = 8.0 atm and is increasing at a rate of 0.10 atm/min and V = 10 F and is decreasing at a rate of 0.15 L/min. Find the rate of change of T with respect to time at that instant if n = 10 moles.

To determine

To find: The rate of change of T with respect to time at that instant if n=10 moles

Explanation

1) Given:

P=8.0 atm

V=10 L

dPdt=0.10 atm/min

dVdt=- 0.15 L/min

Gas law for ideal gas is PV=nRT where n is number of moles and R = 0.0821 is the gas constant

Quotient rule:

ddxfxgx=gxddxfx-fxddx(gx)gx2

Product rule:

ddxfx*gx=fxddxgx+gxddx(fx)

2) Calculation:

Consider the gas law for ideal gas,

PV=nRT

Isolate T,

T=PVnR

Now differentiate with respect to t on both sides,

dTdt=ddt(PVnR)

By using quotient rule,

dTdt=nR ddtPV-PV ddt(nR)

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