Chapter 1, Problem 1.32E

Interpretation Introduction

(a)

Interpretation:

The ideal gas law as volume being a function of pressure and temperature is to be rewritten.

Concept introduction:

The ideal gas law or general gas equation is the equation of state of a hypothetical ideal gas. Thought it has some limitations, it is a good approximation of the behavior of several gases under several conditions. The term was first coined by Emile Clapeyron in the year of 1834 as combination of other laws. The ideal gas law can be written as PV = nRT.

Answer to Problem 1.32E

Volume as a function of pressure can be stated as,

V = F(P) = nRT/P

$\left(\frac{\delta V}{\delta p}\right)={\left[\frac{\delta }{\delta p}\left(\frac{nRTV}{P}\right)\right]}_{T,n}$

Volume s a function of pressure can be stated as,

V = F(T) = nRT/P

$\left(\frac{\delta V}{\delta T}\right)={\left[\frac{\delta }{\delta T}\left(\frac{nRT}{P}\right)\right]}_{T,n}$

Explanation of Solution

The various properties of gases which can be observed with our oral senses, include pressure, temperature, mass and the volume which contains the gas. The careful examination determined that these variables are related to one another and the state of the gas can be determined by the changes of these properties. Boyle’s law is an experimental gas law which describes how the pressure of a gas tends to increase as the volume of the container decreases. Similarly, Charles’s law or law of volumes is an experimental gas law which denotes the expansion of gas when heated.

The ideal gas equation is PV = nRT……………………………….(1)

Where,

P = Pressure of gas

V = Volume of gas

n = No of moles of gas

R = Gas constant and T = Temperature of gas

The gas constant or ideal gas constant (R) is equivalent to the Boltzmann constant and expressed as energy per temperature increment per mole units (R = 0.083 L. bar / K mol).

From the above expression we can know that gas constant ‘R’ is a constant, whose value will not change with respect to any other values but not a variable. The terms pressure, volume and temperature are variables and their values will change with respect to other. They can be expressed with respect to other variables. The slope of a line can be defined as the plane containing the x and y axes and represented by the letter ‘m’. In other words, the change in the y-axis divided by the corresponding change in the x-axis, between two well-defined points on the line. This can be described by the following equation;

$\text{m}\text{=}\frac{\Delta y}{\Delta x}=\frac{\text{rise}}{\text{run}}=\frac{\text{vertical}\text{change}}{\text{horizontal}\text{change}}$

The ideal gas equation (1) can be written as,

(a) Volume as a function of pressure can be stated as,

V = F(P) = nRT/P

$\left(\frac{\delta V}{\delta T}\right)={\left[\frac{\delta }{\delta p}\left(\frac{nRT}{P}\right)\right]}_{T,n}$

Volume s a function of pressure can be stated as,

V = F(T) = nRT/P

$\left(\frac{\delta V}{\delta T}\right)={\left[\frac{\delta }{\delta T}\left(\frac{nRT}{P}\right)\right]}_{T,n}$

Conclusion

Thus, the ideal gas law as volume being a function of pressure and temperature is rewritten.

Interpretation Introduction

(b)

Interpretation:

The expression for the total derivative dV as a function of pressure and temperature is to be stated.

Concept introduction:

The ideal gas law or general gas equation is the equation of state of a hypothetical ideal gas. Thought it has some limitations, it is a good approximation of the behavior of several gases under several conditions. The term was first coined by Emile Clapeyron in the year of 1834 as combination of other laws. The ideal gas law can be written as PV = nRT.

Answer to Problem 1.32E

The expression for the total derivative dV as a function of pressure and temperature is as follows;

total derivative dV as a function of pressure

$\delta V={\left[\frac{-nRT}{P^2}\right]}_{T,n}\delta P$

total derivative dV as a function of temperature

$\delta V={\left[\frac{nR}{P}\right]}_{T,n}\delta T$

Explanation of Solution

The various properties of gases which can be observed with our oral senses, include pressure, temperature, mass and the volume which contains the gas. The careful examination determined that these variables are related to one another and the state of the gas can be determined by the changes of these properties. Boyle’s law is an experimental gas law which describes how the pressure of a gas tends to increase as the volume of the container decreases. Similarly, Charles’s law or law of volumes is an experimental gas law which denotes the expansion of gas when heated.

The ideal gas equation is PV = nRT……………………………….(1)

Where,

P = Pressure of gas

V = Volume of gas

n = No of moles of gas

R = Gas constant and T = Temperature of gas

The gas constant or ideal gas constant (R) is equivalent to the Boltzmann constant and expressed as energy per temperature increment per mole units (R = 0.083 L. bar / K mol).

From the above expression we can know that gas constant ‘R’ is a constant, whose value will not change with respect to any other values but not a variable. The terms pressure, volume and temperature are variables and their values will change with respect to other. They can be expressed with respect to other variables. The slope of a line can be defined as the plane containing the x and y axes and represented by the letter ‘m’. In other words, the change in the y-axis divided by the corresponding change in the x-axis, between two well-defined points on the line. This can be described by the following equation;

$\text{m}\text{=}\frac{\Delta y}{\Delta x}=\frac{\text{rise}}{\text{run}}=\frac{\text{vertical}\text{change}}{\text{horizontal}\text{change}}$

The ideal gas equation (1) can be written as, dV as a function of pressure

$\frac{\delta V}{\delta p}={\left[\frac{\delta }{\delta p}\left(\frac{nRT}{P}\right)\right]}_{T,n}$

$\begin{array}{l}\left(\frac{\delta V}{\delta p}\right)={\left[\frac{-nRT}{P^2}\right]}_{T,n}\\ \delta V={\left[\frac{-nRT}{P^2}\right]}_{T,n}\delta T\end{array}$

Similarly, dV as a function of temperature,

$\frac{\delta V}{\delta T}={\left[\frac{\delta }{\delta T}\left(\frac{nRT}{P}\right)\right]}_{T,n}$

$\begin{array}{l}\left(\frac{\delta V}{\delta T}\right)={\left[\frac{nR}{P}\right]}_{T,n}\\ \delta V={\left[\frac{nR}{P}\right]}_{T,n}\delta T\end{array}$

Conclusion

The expression for the total derivative dV as a function of pressure and temperature is stated.

Interpretation Introduction

(c)

Interpretation:

At a pressure of 1.08 atm and 350 K for one more of ideal gas, the predicted change in volume if the pressure changes by 0.10 atm (that is, dp = 0.10 atm) and the temperature change is 10.0 K is to be calculated.

Concept introduction:

The ideal gas law or general gas equation is the equation of state of a hypothetical ideal gas. Thought it has some limitations, it is a good approximation of the behavior of several gases under several conditions. The term was first coined by Emile Clapeyron in the year of 1834 as combination of other laws. The ideal gas law can be written as PV = nRT.

Answer to Problem 1.32E

The predicted change in volume if the pressure changes by 0.10 atm (dV) = -24.62 liter and the predicted change in volume if the temperature changes by 10.0 K (dV) = 0.76 liter.

Explanation of Solution

We know that; dV as a function of pressure

$\delta V={\left[\frac{-nRT}{P^2}\right]}_{T,n}\delta P$………………………………………(2)

Given;

n = 1 mol ; R = 0.0823 L. atm / K mol

T = 350 K ; P = 1.08 atm

dp = 0.10 atm

substituting the values in equation (2), we get

$\begin{array}{l}\delta V=\frac{-1\text{mol}\times \text{0}\text{.0823}\text{L}\text{.atm/K mol}\times \text{350}\text{K}\times \text{0}\text{.10}\text{atm}}{{(1.08\text{atm)}}^2}\\ =\frac{-28.81}{1.17}\\ \delta V=-24.62\text{liter}\end{array}$

Similarly, we know that dV as a function of temperature

$\delta V={\left[\frac{nR}{P}\right]}_{T,n}\delta T$……………………………..(3)

Given;

n = 1 mol; R = 0.0823 L. atm / K mol

T = 350 K ; P = 1.08 atm

dT = 10 K

substituting the values in equation (3), we get

$\begin{array}{l}\delta V=\frac{1\text{mol}\times \text{0}\text{.0823}\text{L}\text{.atm/K mol}\times \text{10K}}{1.08\text{atm}}\\ =\frac{0.82}{1.08}\text{L}\\ \delta V=0.76\text{liter}\end{array}$

Conclusion

Thus, the predicted change in volume if the pressure changes by 0.10 atm (dV) = -24.62 liter and the predicted change in volume if the temperature changes by 10.0 K (dV) = 0.76 liter is

calculated.