Problem 123 - Chemistry An Atoms First Approach

Question

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 kelvin). a. \(P V\) versus \(V\) with constant \(T\) b. \(P\) versus \(T\) with constant \(V\) 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 \(P\)

Step-by-Step Solution

Answer

Here are the qualitative graphs for the given scenarios: a. PV versus V (Constant T): Horizontal straight line parallel to the V-axis. b. P versus T (Constant V): Straight line that passes through the origin with a positive slope. c. T versus V (Constant P): Straight line that passes through the origin with a positive slope. d. P versus V (Constant T): Hyperbola, passing through the first and third quadrants. e. P versus 1/V (Constant T): Straight line that passes through the origin and has a positive slope. f. PV/T versus P (Constant n, R): Horizontal straight line parallel to the P-axis (x-axis).

Step 1: Qualitative Graph for PV versus V (Constant T)

Since T is constant, multiplying P by V will keep the value constant as well. Therefore, the graph of PV versus V will be a horizontal straight line parallel to the x-axis (V-axis). #Step 2: Graph of P versus T with constant V#

Step 2: Qualitative Graph for P versus T (Constant V)

Using the Ideal Gas Law (PV= nRT), we rearrange the equation to get P=(nR/V)T. Since V is constant and nR is constant, it results in a linear relationship between P and T. The graph of P versus T will be a straight line that passes through the origin and has a positive slope. #Step 3: Graph of T versus V with constant P#

Step 3: Qualitative Graph for T versus V (Constant P)

Rearrange the Ideal Gas Law to solve for T: T= PV/nR. Since P is constant, the relationship between T and V becomes linear. Therefore, the graph of T versus V will be a straight line that passes through the origin and has a positive slope. #Step 4: Graph of P versus V with constant T#

Step 4: Qualitative Graph for P versus V (Constant T)

The Ideal Gas Law can be rewritten as P = nRT/V. With constant T and the product of nR also being constant, there will be an inverse relationship between P and V. In other words, as V increases, P will decrease. The graph of P versus V will be a hyperbola, passing through the first and third quadrants. #Step 5: Graph of P versus 1/V with constant T#

Step 5: Qualitative Graph for P versus 1/V (Constant T)

Rearrange the Ideal Gas Law to get P=nRT/V. By replacing 1/V with X, we can see that the relationship between P and 1/V is linear (since nRT is constant), with P = nRTX. The graph of P versus 1/V will be a straight line passing through the origin and has a positive slope. #Step 6: Graph of PV/T versus P#

Step 6: Qualitative Graph for PV/T versus P (Constant n, R)

From the Ideal Gas Law, we know that PV=nRT. Dividing both sides of the equation by T gives PV/T = nR. Since n and R are constants, the graph of PV/T versus P will also be a horizontal straight line parallel to the P-axis (x-axis).