Respuesta :
Answer:
(a) [tex]\frac{dP}{dV}=\frac{2an^2}{V^3}-\frac{nRT}{(V-nb)^2}[/tex]
(b) [tex]\frac{dP}{dt}=\frac{nR(V-nb)\frac{dT}{dt}-nRT\frac{dV}{dt}}{(V-nb)^2}+\frac{2an^2\frac{dV}{dt}}{V^3}[/tex]
(c) [tex]\frac{dP}{dV}[/tex] is the rate change of pressure of the gas per unit volume.
[tex]\frac{dP}{dt}[/tex] is the rate change of pressure of the gas per unit time.
Step-by-step explanation:
Formula:
- [tex]\frac{d}{dx}(\frac uv)=\frac {v\frac{d}{dx}v-u\frac{d}{dx}v}{v^2}[/tex]
Given that,
[tex]P=\frac{nRT}{V-nb}-\frac{an^2}{V^2}[/tex]
(a)
[tex]P=\frac{nRT}{V-nb}-\frac{an^2}{V^2}[/tex]
Differentiating with respect to V
[tex]\frac{dP}{dV}=\frac{(V-nb)\frac{d}{dV}(nRT)-(nRT)\frac{d}{dV}(V-nb)}{(V-nb)^2}-\frac{V^2\frac{d}{dV}(an^2)-(an^2)\frac{d}{dV}V^2}{(V^2)^2}[/tex]
[tex]=\frac {-nRT}{(V-nb)^2}-\frac{-an^2. 2V}{V^4}[/tex]
[tex]=\frac{2an^2}{V^3}-\frac{nRT}{(V-nb)^2}[/tex]
(b)
[tex]P=\frac{nRT}{V-nb}-\frac{an^2}{V^2}[/tex]
Differentiating with respect to t
[tex]\frac{dP}{dt}=\frac{(V-nb)\frac{d}{dt}(nRT)-(nRT)\frac{d}{dt}(V-nb)}{(V-nb)^2}-\frac{V^2\frac{d}{dt}(an^2)-(an^2)\frac{d}{dt}V^2}{(V^2)^2}[/tex]
[tex]=\frac{(V-nb)nR\frac{dT}{dt}-nRT\frac{dV}{dt}}{(V-nb)^2}-\frac{-an^2.2V\frac{dV}{dt}}{V^4}[/tex]
[tex]=\frac{nR(V-nb)\frac{dT}{dt}-nRT\frac{dV}{dt}}{(V-nb)^2}+\frac{2an^2\frac{dV}{dt}}{V^3}[/tex]
(c)
[tex]\frac{dP}{dV}[/tex] is the rate change of pressure of the gas per unit volume.
[tex]\frac{dP}{dt}[/tex] is the rate change of pressure of the gas per unit time.
