Chapter 5: Problem 123
Draw a qualitative graph to show how the first property varies with the second in each of the following (assume 1 mole of an ideal gas and \(T\) in kelvins). a. \(P V\) versus \(V\) with constant \(T\) b. \(P\) versus \(T\) with constant \(\underline{V}\) c. \(T\) versus \(V\) with constant \(P\) d. \(P\) versus \(V\) with constant \(T\) e. \(P\) versus \(1 / V\) with constant \(T\) f. \(P V / T\) versus \(P\)
Short Answer
Expert verified
In summary, the qualitative graphs for the given relationships are as follows:
a. PV vs V (constant T) will yield a horizontal line at the constant value of RT.
b. P vs T (constant V) will generate a straight line passing through the origin with a slope of nR/V.
c. T vs V (constant P) will give a straight line passing through the origin with a slope of P/nR.
d. P vs V (constant T) will result in a hyperbola with asymptotes at P = 0 and V = 0.
e. P vs 1/V (constant T) will create a straight line passing through the origin with a slope of nRT.
f. PV/T vs P will display a horizontal line at the constant value of nR.
Step by step solution
01
- Write down the ideal gas law equation
The ideal gas law equation is given by: \(PV = nRT\), where \(n = 1\) since we are given 1 mole of an ideal gas.
02
- Isolate the product (PV)
Rewrite the ideal gas law equation as: \(PV = RT\)
03
- Draw the graph
Since the temperature (T) is constant, \(RT\) is also a constant value. Therefore, the equation becomes \(y = k\), where \(y = PV\) and \(k\) is a constant value. The graph will be a horizontal line at the constant value of \(RT\).
b. P versus T with constant V
04
- Write down the ideal gas law equation
The ideal gas law equation is given by: \(PV = nRT\), where \(n = 1\) since we are given 1 mole of an ideal gas.
05
- Isolate the pressure (P)
Rewrite the ideal gas law equation as: \(P = \frac{nRT}{V}\)
06
- Draw the graph
Since the volume (V) is constant, the equation becomes \(y = mx\), where \(y = P\) and \(m = \frac{nR}{V}\). The graph will be a straight line passing through the origin with a slope of \(\frac{nR}{V}\).
c. T versus V with constant P
07
- Write down the ideal gas law equation
The ideal gas law equation is given by: \(PV = nRT\), where \(n = 1\) since we are given 1 mole of an ideal gas.
08
- Isolate the temperature (T)
Rewrite the ideal gas law equation as: \(T = \frac{PV}{nR}\)
09
- Draw the graph
Since the pressure (P) is constant, the equation becomes \(y = mx\), where \(y = T\) and \(m = \frac{P}{nR}\). The graph will be a straight line passing through the origin with a slope of \(\frac{P}{nR}\).
d. P versus V with constant T
10
- Write down the ideal gas law equation
The ideal gas law equation is given by: \(PV = nRT\), where \(n = 1\) since we are given 1 mole of an ideal gas.
11
- Isolate the pressure (P)
Rewrite the ideal gas law equation as: \(P = \frac{nRT}{V}\)
12
- Draw the graph
Since the temperature (T) is constant, the equation becomes \(y = \frac{k}{x}\), where \(y = P\) and \(k = nRT\). The graph will be a hyperbola with the horizontal asymptote at the x-axis (P = 0) and the vertical asymptote at the y-axis (V = 0).
e. P versus 1/V with constant T
13
- Write down the ideal gas law equation
The ideal gas law equation is given by: \(PV = nRT\), where \(n = 1\) since we are given 1 mole of an ideal gas.
14
- Apply the inverse relationship (P versus 1/V)
Rewrite the ideal gas law equation as: \(P = \frac{nRT}{V}\)
15
- Draw the graph
Since the temperature (T) is constant, the equation becomes \(y = mx\), where \(y = P\) and \(x = \frac{1}{V}\), and \(m = nRT\). The graph will be a straight line passing through the origin with a slope of \(nRT\).
f. PV/T versus P
16
- Write down the ideal gas law equation
The ideal gas law equation is given by: \(PV = nRT\), where \(n = 1\) since we are given 1 mole of an ideal gas.
17
- Isolate PV/T
Rewrite the ideal gas law equation as: \(PV = nRT\), then divide both sides by (T): \(\frac{PV}{T} = nR\)
18
- Draw the graph
Since the number of moles (n) and the gas constant (R) are constants, the equation becomes \(y = k\), where \(y = \frac{PV}{T}\) and \(k = nR\). The graph will be a horizontal line at the constant value of \(nR\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
PV versus V graph at constant T
To understand the relationship between the product of pressure and volume (PV) and volume (V) at constant temperature, let's first recall the ideal gas law, which states that for one mole of an ideal gas, PV equals nRT. With temperature (T) being constant and nR as well, PV remains constant regardless of changes in V.
Hence on a graph where PV is plotted on the y-axis and V on the x-axis, you would see a perfectly horizontal line that reflects this constant relationship. The value along the y-axis will be equal to RT for our graph, with R being the gas constant and T being the consistent temperature.
Hence on a graph where PV is plotted on the y-axis and V on the x-axis, you would see a perfectly horizontal line that reflects this constant relationship. The value along the y-axis will be equal to RT for our graph, with R being the gas constant and T being the consistent temperature.
P versus T graph at constant V
For a graph that showcases the pressure (P) of an ideal gas against its temperature (T) at a constant volume (V), the ideal gas law again comes into play. Under these conditions, the equation reduces to P being directly proportional to T, as V and n are held steady.
The resultant graph is a straight line initiating from the origin with a slope that equals nR/V. This slope indicates the rate at which the pressure increases compared to the temperature, visually emphasizing the direct relationship between these two variables when volume is fixed.
The resultant graph is a straight line initiating from the origin with a slope that equals nR/V. This slope indicates the rate at which the pressure increases compared to the temperature, visually emphasizing the direct relationship between these two variables when volume is fixed.
T versus V graph at constant P
When we examine how temperature (T) varies with volume (V) at a constant pressure (P), we once more tap into the ideal gas equation. This time, our focus pins down P and n, leaving T and V to be dependent upon each other. Remarkably, they are directly proportional, illustrated by the fact that increasing the volume at constant pressure will lead a gas to absorb heat and rise in temperature.
The graph renders a straight line rising from the origin whose angle is depicted through the slope, P/nR. This slope guides us to see how temperature escalates as volume is expanded while pressure is conserved.
The graph renders a straight line rising from the origin whose angle is depicted through the slope, P/nR. This slope guides us to see how temperature escalates as volume is expanded while pressure is conserved.
P versus V graph at constant T
Pressure (P) versus volume (V) graphs under a blanket of unchanging temperature (T), familiarly known as isotherms, are a classic representation of Boyle's Law within the ideal gas law framework. The law states that at a set temperature, the pressure of a gas is inversely proportional to its volume.
In terms of graphing, this inverse relationship equates to a hyperbolic curve that never quite touches the axes, denoting that neither P nor V will reach zero. Reducing the volume results in increased pressure, creating a steep downtrend on the left side of the graph, while expanding the volume decreases pressure, manifesting as a gentle slope on the right.
In terms of graphing, this inverse relationship equates to a hyperbolic curve that never quite touches the axes, denoting that neither P nor V will reach zero. Reducing the volume results in increased pressure, creating a steep downtrend on the left side of the graph, while expanding the volume decreases pressure, manifesting as a gentle slope on the right.
P versus 1/V graph at constant T
When we flip the scenario and examine the relationship between pressure (P) and the inverse of volume (1/V) at steady temperature, the ideal gas law betrays yet another linear relationship. This is due to P being directly proportional to 1/V when the temperature is a constant factor.
Placing P on the y-axis and 1/V on the x-axis gives rise to a straight line that crosses through the origin. The slope of this line is the product nRT, which articulates how pressure will escalate as the space a gas occupies is reduced.
Placing P on the y-axis and 1/V on the x-axis gives rise to a straight line that crosses through the origin. The slope of this line is the product nRT, which articulates how pressure will escalate as the space a gas occupies is reduced.
PV/T versus P graph
Lastly, examining PV/T versus P. According to the ideal gas law, when we isolate PV/T for a single mole of an ideal gas, it simplifies to the gas constant, R. This indicates that PV/T is also a constant, which remains unaltered regardless of pressure variations.
Graphing this brings about a horizontal line that cuts across the y-axis at nR. Regardless of any movements along the x-axis, which represents pressure, the ratio of PV to T remains constant, solidifying the idea that for an ideal gas, the change in pressure does not affect this proportion at a given temperature.
Graphing this brings about a horizontal line that cuts across the y-axis at nR. Regardless of any movements along the x-axis, which represents pressure, the ratio of PV to T remains constant, solidifying the idea that for an ideal gas, the change in pressure does not affect this proportion at a given temperature.
