Differential

Question

The ideal gas law states that for an ideal gas, PV =nRT,
where
P = pressure of the gas, V = volume of the gas,
n = number of moles of the gas,
R = universal gas constant,
T = temperature of the gas in degree Kelvins.
Suppose there’s a sample of a gas in a chamber and this gas is being compressed by a piston. During this process, it undergoes a change in volume, pressure as well as temperature. Assuming this gas behaves ideally, use the rules of differ- entiation to derive the rate of change of volume V of the chamber w.r.t time t in termsof dP , dT ,P,T,nandR.

Answer 1

Hello Gerald,

Each of the 3 quantities, V, P & T are time dependent i.e they are functions of time and we will treat them as such.

Now we will rewrite the equation in a form that clearly expresses this.

 

V(t) . P(t) = nR T(t) {nR is a constant}

Now differentiate both sides recognizing that the LHS is a product of two functions of time.

 

Thus V (dP/dt) + P(dV/dt) = nR(dT/dt)

 

Rearranging terms (dV/dt) = (1/P)[nR(dT/dt) - V(dP/dt)]

now substitute for V in this equation to complete the answer. ( V = nRT/P)

 

Hope this helps.

Wishing you the best,

Pradip