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. $PV$ 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. $PV/T$ versus $P$

Short 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-by-step solution

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#

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#

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#

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#

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#

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).