(a) Use the ideal gas law to show that, for an ideal gas at...

(a) Use the ideal gas law to show that, for an ideal gas at constant pressure, the coefficient of volume expansion is equal to $\beta=1 / T,$ where $T$ is the temperature in kelvins. Compare to Table $13-1$ for gases at $T=293 \mathrm{K}$ . $(b)$ Show that the bulk modulus (Section $9-5$ ) for an ideal gas held at constant temperature is $B=P,$ where $P$ is the pressure.

(a) Use the ideal gas law to show that, for an ideal gas at constant pressure, the coefficient of volume expansion is equal to $\beta=1 / T,$ where $T$ is the temperature in kelvins. Compare to Table $13-1$ for gases at $T=293 \mathrm{K}$ . $(b)$ Show that the bulk modulus (Section $9-5$ ) for an ideal gas held at
constant temperature is $B=P,$ where $P$ is the pressure.

Key Concepts

Coefficient of Volume Expansion

The coefficient of volume expansion is defined as ? = (1/V)(?V/?T)_p, which quantifies how much the volume of a substance changes with temperature at constant pressure. For an ideal gas, this concept directly ties back to the ideal gas law, showing that ? is inversely proportional to the absolute temperature. This relationship is an important characteristic of materials and ideal gases, highlighting their expansion behavior when heated or cooled.

Bulk Modulus

The bulk modulus is a measure of a material’s resistance to uniform compression, defined as B = -V(dP/dV). In the context of an ideal gas undergoing an isothermal process, the bulk modulus becomes equivalent to the pressure P. This simplification is significant in thermodynamic problems, as it connects the mechanical property of compressibility directly with the measurable pressure of the gas, illustrating how changes in volume impact the system's pressure under constant temperature.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates the pressure, volume, and temperature of an ideal gas. It serves as a foundational relation in thermodynamics, allowing one to derive various thermodynamic properties. By assuming an ideal behavior where intermolecular forces are negligible and the volume of individual gas molecules is insignificant compared to the container’s volume, this law simplifies the analysis of gas systems under different conditions, such as constant pressure or temperature.

Transcript

00:01 In the first item of this question, we have to use the ideal gas law to show that the thermal expansion coefficient is given by 1 over t when the gas is held at a constant pressure.

00:12 So we begin by using the definition of that beta coefficient.

00:17 The definition for that coefficient is the following.

00:20 Beta is calculated by delta v divided by v0 times 1 over delta t.

00:30 And now we do the following.

00:32 Delta v is nothing else than the new volume, v prime, minus the old volume that i'll call v.

00:39 And this is divided by the old volume v, multiplied by 1 over delta t.

00:45 Okay, then we can write this as v.

00:49 V -prime divided by v minus 1, and this quantity is multiplying 1 over delta -t.

00:57 To calculate v prime, we can use the ideal gas law.

01:01 The ideal gas law tells us that the volume times the pressure is given by the number of molecules times k times the temperature t.

01:11 Now notice that k is a constant, it will never change.

01:16 So we can say that pressure times volume divided by number of molecules times temperature is equals to a constant.

01:24 And this is through both before and after some transformation.

01:28 So p -prime, v -prime divided by n -t prime t -prime is still equal to k.

01:34 And with that, we derived an expression which says the following.

01:38 P -t times v divided by n times t is equal to p -prime times v -prime divided by n -prime times t -prime.

01:46 Now that you have this expression, let me erase the derivation because we won't need that anymore.

01:52 Okay, let me drag this.

01:54 It here.

01:55 So applying to the situation of the first item, we get the following.

01:59 P times v divided by n times t is equal to p prime, v.

02:06 Divided by n prime times t prime.

02:09 But in that situation, it's true that the pressure is kept constant.

02:13 So p is equal to p prime as the same is true for the number of molecules.

02:17 So n is equal to n prime.

02:19 Then you can simplify p and p prime and n with n prime.

02:25 4 we are left with the following.

02:27 V over t is equal to v prime over t prime.

02:31 Then the new volume v prime is equal to v times t prime divided by t.

02:39 Then using this result in the expression for beta, we get the following.

02:45 Beta is equal to v prime, which is v times t prime divided by t divided by v...

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