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For $P=n R T / V$, find $\frac{\partial}{\partial V} P$ and $\frac{\partial}{\partial T} P$. For fixed $T$, how does $P$ change as $V$ increases? For fixed $V,$ how does $P$ change as $T$ increases?

Calculus 3

Chapter 13

Two Variable Calculus and Diffusion

Section 1

Partial derivatives of functions of two variables

Partial Derivatives

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Lectures

12:15

In calculus, partial deriv…

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Prove that $$\left(\frac{\…

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The function $P(T, V)=\fra…

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$\text { Show that } c_{p}…

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Let $f(p, q)=1-p(1+q) .$ F…

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If $p^{3}+s q=t,$ and $q^{…

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Let $z=f(x, y), x=g(s, t),…

01:27

02:03

Suppose $p$ varies directl…

03:00

Show that $$c_{p, g}-c_{p …

04:20

The gas law for a fixed ma…

03:12

01:49

Suppose that $x=g(t, s), y…

00:27

Calculate $\partial P / \p…

01:34

Work each problem.Suppose …

02:26

Show that $$c_{v}=-T\left(…

02:39

Show that $$\beta=\alpha(\…

03:02

An ideal gas goes through …

04:21

Demonstrate that $$k=\frac…

03:04

$$\begin{array}{l}{\te…

02:42

Assume that $w=f\left(t s^…

in this question. We need to show this relationship. Okay, So the equations that we need to use the maximal relations Hey in particular, uh, the left hand side pressure p pressure T He s at six s is equal to one over. I should t pasha peed. That makes us using the reciprocity relation. And, uh, the nominator is the one that we use our actual relation. So this is partial V partial s at six p. And so this is, uh, pressure as partial v at six p. Okay. And then, um and also specialty partial V at 60 is equal to partial s partial B at 16. Okay. Okay, then another relationship that we need eyes about. Okay, so things k, it goes to CP over c v. Okay. It's okay. OK. Minus one is equal to C. P or T V. You have my cp over CV minus one. So we have CP given by CP minus C B. Hey, the next thing we need is, uh, definitions off. C p N C v uh huh. Definitions off CP and C P minus E v. Hey, this can be found from the book, so B p is equal through, uh, tee times partial s specialty at 60. So this is equal to using change rule. You have partial as partial B at 16 and partial v partial t at six p Hey and using Maxwell relationship acceleration The pressure as partial B is equal to partial P casualty that fix us and then multiplied by partial v partial t A 16 and then CPM on TV is equal toe minus t partial v partial T at six p square Parents partial p partial V at 16. Teen now we can get a order pieces. Yes, with the right hand side. The sequel to K OK, minus one. Uh, p partial T RCMP pasha t fix me. This is equal to CP Divide by C. P minus E v on special P specialty at fixed B Then this substitute C p is equal toe tee times partial p partial T at fixed s times Partial re passion 30 at six p. Good bye Bye. Minus t I shall be partial. T at 60 square Pasha p slash B at 60. Multiplied by partial P partial T at 60. Okay, now we can see that one off the partial d reduce cancer out and the temperature so cancer out. Okay, so we have minus one times partial p partial t that fix us. Uh, and partial P pasha t that fixed V and the derivatives and the partial derivatives in the denominator, we use the reciprocity relationship. Okay, so partial v partial t get specialty, partial VFX p. And then similarly, get pressure. Be partial. T a 16. Okay, so, uh, it's chickens or here. Okay, this is equal to minus one because of the secret relationship. Now, the secret property off partial derivatives. Okay, so this is equal to minus one. So we have pressure p pressure t that takes us, which is the left hand side. Okay, so we managed to show that Brian Side goes to nothing site and so sure, and that's all

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In calculus, partial derivatives are derivatives of a function with respect …