DATA You compress a gas in an insulated cylinder-no heat...

DATA You compress a gas in an insulated cylinder-no heat flows into or out of the gas. The gas pressure is fairly low, so treating the gas as ideal is a good approximation. When you measure the pressure as a function of the volume of the gas, you obtain these results: $$ \begin{array}{l|lllll} \boldsymbol{V}(\mathbf{L}) & 2.50 & 2.02 & 1.48 & 1.01 & 0.50 \\ \hline \boldsymbol{p}(\mathbf{a t m}) & 0.101 & 0.139 & 0.202 & 0.361 & 0.952 \end{array} $$ (a) Graph $\log (p)$ versus $\log (V),$ with $p$ in $\mathrm{Pa}$ and $V$ in $\mathrm{m}^{3} .$ Explain why the data points fall close to a straight line. (b) Use your graph to calculate $\gamma$ for the gas. Is the gas monatomic, diatomic, or polyatomic? (c) When $p=0.101$ atm and $V=2.50 \mathrm{~L}$, the temperature is $22.0^{\circ} \mathrm{C}$. Apply the ideal-gas equation and calculate the temperature for each of the other pairs of $p$ and $V$ values. In this compression, does the temperature of the gas increase, decrease, or stay constant?

DATA You compress a gas in an insulated cylinder-no heat flows into or out of the gas. The gas pressure is fairly low, so treating the gas as ideal is a good approximation. When you measure the pressure as a function of the volume of the gas, you obtain these results:
$$
\begin{array}{l|lllll}
\boldsymbol{V}(\mathbf{L}) & 2.50 & 2.02 & 1.48 & 1.01 & 0.50 \\
\hline \boldsymbol{p}(\mathbf{a t m}) & 0.101 & 0.139 & 0.202 & 0.361 & 0.952
\end{array}
$$
(a) Graph $\log (p)$ versus $\log (V),$ with $p$ in $\mathrm{Pa}$ and $V$ in $\mathrm{m}^{3} .$ Explain why the data points fall close to a straight line. (b) Use your graph to calculate $\gamma$ for the gas. Is the gas monatomic, diatomic, or polyatomic? (c) When $p=0.101$ atm and $V=2.50 \mathrm{~L}$, the temperature is $22.0^{\circ} \mathrm{C}$. Apply the ideal-gas equation and calculate the temperature for each of the other pairs of $p$ and $V$ values. In this compression, does the temperature of the gas increase, decrease, or stay constant?

Key Concepts

Ideal Gas Law

The ideal gas law (PV = nRT) relates the pressure, volume, and temperature of an ideal gas. It serves as a fundamental principle in thermodynamics that allows one to calculate any one of these properties provided the other two are known. This law is widely applicable for gases at relatively low pressures and high temperatures, where gas molecules exhibit negligible interactions.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the environment. In such a process, any change in the internal energy of the gas is exclusively due to work done on or by the system. For an ideal gas undergoing an adiabatic process, the pressure and volume are related by the equation PV^? = constant, where ? is the specific heat ratio.

Specific Heat Ratio (?)

The specific heat ratio, ?, is defined as the ratio of the heat capacities at constant pressure and constant volume (? = C_p/C_v). It is a crucial parameter in characterizing the thermodynamic behavior of gases during adiabatic processes. The value of ? provides insight into the molecular composition of the gas, with different values typically corresponding to monatomic, diatomic, or polyatomic gases.

Log-Log Graphing for Power Law Relationships

Graphing the logarithm of two variables can linearize power law relationships. When a relation is of the form y = kx^n, taking the logarithm of both sides results in log(y) = n*log(x) + log(k), which is a straight line with slope n. This method is particularly useful in experimental thermodynamics for determining exponents in equations such as PV^? = constant.

Temperature Changes in Adiabatic Compression

In an adiabatic compression, since there is no heat transfer, the work done on the gas results in an increase in its internal energy, thereby increasing the temperature. This concept underscores the energy conservation principle in thermodynamics and is analyzed using the ideal gas law along with the adiabatic condition.

Transcript

00:01 In this problem, we're going to talk about the ideal gas law and adiabatic processes.

00:05 And what we need to remember is that for an ideal gas, the pressure times the volume is equal to the number of moles times the ideal gas constant times the temperature.

00:17 Also we need to remember that for an adabatic process, p times v to the gamma is constant throughout the process.

00:27 K here is just a constant.

00:29 They not confuse it with the boatsman constant.

00:31 It's just a general constant for each process.

00:35 Okay, for each process has its own value of k, where gamma is the adabetic index, and it's equal to the molar heat capacity at constant pressure divided by the molar heat capacity at constant volume.

00:53 Okay, so what we have in our problem is this table that shows the volume as a function of the pressure for a gas that undergoes an adiabatic process.

01:11 And what we have to do is to calculate the logarithm of v and the logarithm of p where v is in cubic meters and p in pascal.

01:28 And then we have to plot the graph and say why we explain, why we expect this graph to be a straight line.

01:39 Okay, so let's start.

01:41 Basically what i'm going to write here in the table, what i'm going to add to this table is are the values of the logarithms.

01:51 So the logarithm of 2 .5, notice that i want to calculate the logarithm of 2 .5 times sent to the minus 3 because each letter is sent to the minus 3 cubic meters.

02:05 So log of v is minus 6, log of 2 .02 times into the minus 3 is minus 6 .2, then log of 1 .48, it's minus 6 .52.

02:22 The next one is minus 6 .9.

02:25 The next one is minus 7 .6.

02:29 The log of p, then we're going to calculate the log of 0 .101 times 1 .1 times 10 to the fifth, scals because i'm transforming from atmospheres to the skull.

02:46 So the first log is equal to 9 .23.

02:51 The second one is 9 .55.

02:54 The third is 9 .92.

02:59 The fourth is 10 .5.

03:02 And the fifth is 11 .47.

03:07 And now we have to graph this.

03:11 And i'm going to put in the y -axis, i'm going to put log of p.

03:18 In the x -axis, i'm going to put log of v.

03:24 And i'm just going to, actually, it would be a little better to do it a little differently.

03:32 So let me notice that the log of v is generally negative.

03:36 So i'm going to consider that.

03:40 I'll take that into account.