Final answer:
To find dp/dt in the ideal gas law when both volume and temperature are changing in time, we need to use the chain rule. Differentiating both sides of the equation with respect to time and rearranging, we get an equation for dp/dt in terms of the given variables.
Step-by-step explanation:
The question asks to find ∂p/∂t assuming that both volume and temperature are changing in time. In order to do this, we need to use the chain rule from calculus. Let's start with the ideal gas law: pV = nRT, where p is pressure, V is volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.
Next, we need to differentiate both sides of the equation with respect to time (t):
d(pV)/dt = d(nRT)/dt
Using the product rule, we can differentiate the left side to obtain:
V(dp/dt) + p(dV/dt) = R(dn/dt)T + nR(dT/dt)
Since we are assuming that both volume (V) and temperature (T) are changing in time, we have dV/dt and dT/dt. Therefore, we can rearrange the equation to solve for dp/dt:
dp/dt = (R/n)(dn/dt - (pV/T)(dT/dt + (V/T)(dp/dt)))