Consider the expansion of an ideal gas at constant pressure....

Consider the expansion of an ideal gas at constant pressure. The initial temperature is $T_{0}$ and the initial volume is $V_{0} .$ (a) Show that $\Delta V / V_{0}=\beta \Delta T,$ where $\beta=1 / T_{0}$ (b) Compare the coefficient of volume expansion $\beta$ for an ideal gas at $20^{\circ} \mathrm{C}$ to the values for liquids and gases listed in Table 13.3

Consider the expansion of an ideal gas at constant pressure. The initial temperature is $T_{0}$ and the initial volume is $V_{0} .$ (a) Show that $\Delta V / V_{0}=\beta \Delta T,$ where $\beta=1 / T_{0}$
(b) Compare the coefficient of volume expansion $\beta$ for an ideal gas at $20^{\circ} \mathrm{C}$ to the values for liquids and gases listed in Table 13.3

Key Concepts

Constant Pressure Process

A constant pressure process implies that during heating or cooling, the pressure remains unchanged, which simplifies the relationship between temperature and volume. In the context of an ideal gas, this condition allows us to isolate the effect of temperature on volume, leading to a direct proportionality that is central to deriving the coefficient of expansion.

Coefficient of Volume Expansion

The coefficient of volume expansion quantifies how much the volume of a substance changes per unit change in temperature, typically defined as the fractional change in volume per degree temperature change. For an ideal gas at constant pressure, this coefficient is derived from the ideal gas law and is given by the reciprocal of the absolute temperature, highlighting how sensitive the gas volume is to temperature variations.

Thermal Expansion

Thermal expansion refers to the tendency of matter to change its volume in response to a change in temperature. For gases, this expansion is often more pronounced than in liquids or solids, and it is directly linked to the kinetic energy of the molecules when heated.

Ideal Gas Law

The ideal gas law, expressed as PV = nRT, relates pressure, volume, and temperature in a simple proportionality, serving as the foundation for understanding the behavior of gases. In problems involving gas expansion or compression at constant pressure, this law directly relates changes in volume to changes in temperature, assuming the quantity of gas remains constant.

Transcript

00:01 In the first item of this question, we have to calculate the variation in the volume divided by the initial volume and show that it can be written in a specific form.

00:11 So we begin this question by expanding this expression.

00:16 So the variation in the volume is the final volume minus the initial volume.

00:22 So that's it.

00:24 Now we can calculate what is the final volume as a function of the initial volume by using that expression that says that the initial pressure times the initial volume divided by the initial temperature is equal to the final pressure times the final volume divided by the final temperature.

00:41 As the problem says, the final pressure is equal to the initial pressure, so we can simplify them.

00:48 Like that.

00:49 Then we can solve this equation for the final volume by sending the final temperature to the other side.

00:55 This will result in a final volume that is given.

00:58 By the initial volume times the final temperature divided by the initial temperature.

01:05 Then we can substitute this value in the equation that we are working on to get v is now v0 times t divided by t0 minus v0 divided by v0.

01:23 We can factor v0 in the numerator to got v0 times t0.

01:29 Divided by t0 minus 1 divided by v0.

01:35 Then we can simplify the volumes v0 with v0...

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