From 0e85074859cf4e2baaeb8d1da306eabf13082412 Mon Sep 17 00:00:00 2001 From: Nathan Nguyen <159226326+Indigo5684@users.noreply.github.com> Date: Thu, 23 Jan 2025 12:24:09 -0600 Subject: [PATCH] Thermal Physics Reading Assignment 3 --- docs/physics/thermal/1-energy.md | 28 ++++++++++++++++++ .../Reading Assignment 3.rnote | Bin 0 -> 376435 bytes 2 files changed, 28 insertions(+) create mode 100644 statics/physics/thermal/reading-assignments/Reading Assignment 3.rnote diff --git a/docs/physics/thermal/1-energy.md b/docs/physics/thermal/1-energy.md index 0b8653d..2648075 100644 --- a/docs/physics/thermal/1-energy.md +++ b/docs/physics/thermal/1-energy.md @@ -139,3 +139,31 @@ Consider a piston with a compressable gas. In tthis case, with $\Delta x$ positi We know that the force exerted by the piston is equal to the pressure times the area. Thus, $W = P A \Delta x$. But $A\Delta x$ is just volume (in this case, negative, as volume is decreasing). So, $W = -P \Delta V$. However, this assumes constant pressure. For a non-constant perssure, $P(V)$, we know that $W = \int F dx = - \int P(V) dV$. + +## Section 1.6 - Heat Capacity + +**Definition**. The *heat capacity* $C = Q / \Delta T$ of an object is the energy required to raise its temperaature by one degree. + +**Definition**. The *specific heat capacity* $c = C / m = C / (m \Delta T)$ is the energy needed to raise a substance by one degree per unit mass. + +In a system, we can note thaht + +$$$C = \frac{Q}{\Delta T} = \frac{\Delta U - W}{\Delta T}$$ + +Consider the case in which volume is constant. Under these circumstances, we can calculate the *heat capacity at constant volume*, and as volume does not change, $W = 0$, so + +$$C_V = (\frac{\Delta U}{\Delta T})_V = (\frac{\partial U}{\partial T})_V$$ + +In the case where pressure is constant instead, we see that the *heat capacity at constant pressurue* is defined as + +$$C_P = (\frac{\Delta U}{\Delta T})_V = (\frac{\Delta U - (- P \Delta V)}{\Delta T})_P = (\frac{\partial U}{\partial T})_P + P (\frac{\partial V}{\partial T})_P$$ + +Knowing that $U = \frac{1}{2}NfkT$, we can see that $C_V = \frac{NfK}{2}$. + +**Theorem**. The *Rule of Dulong and Pitt* states that in solids, $f = 6$, so the heat capacity per mole is $3R$. In reality, all degrees of freedom freeze out as $T$ approaches $0$, so $C$ approaches $0$. + +For an ideal gas, at constant pressurre, + +$$(\frac{\partial V}{\partial T})_P = \frac{\partial}{\partial T} (\frac{NkT}{P}) = \frac{Nk}{P}$$ + +So, $C_P = C_V + Nk = C_V + nR$. diff --git a/statics/physics/thermal/reading-assignments/Reading Assignment 3.rnote b/statics/physics/thermal/reading-assignments/Reading Assignment 3.rnote new file mode 100644 index 0000000000000000000000000000000000000000..0fb145f291487c27f126a9dcf3649bf8a6f65437 GIT binary patch literal 376435 zcmdqI_g52L_~>g#YzT;`C{aMF7!@_r5|Jj&h%}`|q>B&*5eO+NDk36XkdmNu5Ks_M zfkc#!5UHUEks2VhRMK&=cfM|;^n4W5%S&r zwS3rc1$%UHvOn7Y%*gGnTa+Tmmu8M#JieiO97Mh(7MdB?+)PiaK4LYuFVyp6Q$oSr z>bTdLeL^yZN3O&Jus9(DBDNJogiaG2!~zPgssZvy%*BcM#QGZHuNt_RFTA%g&2Q!= z&f@S`3^$6m@>+~x5XAyWSnlYexQ?POW)Lfc7Ys)~6a0!K=d39bKuRni&N6@hUdCt? zQ$!46bNw&jR{q3bOUpe9A&kK;oa7EIp3bl)Ok#8RYi8`B&bV755#j-dD1OKxP+Vdk 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