A group of students is studying a model of gas behavior in a closed cylinder. The model is… — ACT

A group of students is studying a model of gas behavior in a closed cylinder. The model is based on the Ideal Gas Law:

PV = nRT

where $P$ is pressure, $V$ is volume, $n$ is the number of moles of gas, $T$ is temperature, and $R$ is the gas constant. Match each set of theoretical modifications to its resulting effect on the gas variables.

Doubling the volume ( $V$ ) while keeping the temperature ( $T$ ) and number of moles ( $n$ ) constant.The pressure ( $P$ ) is halved: $P_f = 0.5P_i$
Doubling the temperature ( $T$ ) and halving the volume ( $V$ ) while keeping the number of moles ( $n$ ) constant.The pressure ( $P$ ) quadruples: $P_f = 4P_i$
Tripling the number of moles ( $n$ ) and doubling the volume ( $V$ ) while keeping the temperature ( $T$ ) constant.The pressure ( $P$ ) increases by a factor of 1.5: $P_f = 1.5P_i$
Doubling the temperature ( $T$ ) and doubling the pressure ( $P$ ) while keeping the number of moles ( $n$ ) constant.The volume ( $V$ ) remains unchanged: $V_f = V_i$

Answer

Doubling the volume with constant temperature and moles halves the pressure; doubling temperature and halving volume quadruples the pressure; tripling moles and doubling volume increases the pressure by a factor of 1.5; doubling temperature and pressure leaves the volume unchanged.

The correct matches represent mathematically precise rearrangements and scaling of the Ideal Gas Law equation (

PV = nRT

). Doubling

V

decreases

P

to half; doubling

T

while halving

V

compounds to a fourfold increase in

P

; tripling

n

while doubling

V

scales

P

1.5

; and doubling both

T

and

P

leaves

V

constant as the factors cancel each other out.

Step-by-Step Solution

Analyze the relationship for pressure under constant moles and temperature.

Pressure is inversely proportional to volume (

P \propto \frac{1}{V}

). Doubling the volume results in halving the pressure.

To determine the direct effect of volume changes on pressure using the model equation

P = \frac{nRT}{V}

Analyze the combined effect of temperature and volume changes on pressure.

Pressure is directly proportional to temperature and inversely proportional to volume (

P \propto \frac{T}{V}

). Doubling temperature and halving volume increases pressure by a factor of

\frac{2}{0.5} = 4

To calculate the net scaling factor of pressure when two independent variables in the model are modified simultaneously.

Analyze the combined effect of mole and volume changes on pressure.

Pressure is directly proportional to the number of moles and inversely proportional to volume (

P \propto \frac{n}{V}

). Tripling the moles and doubling the volume increases pressure by a factor of

\frac{3}{2} = 1.5

To evaluate the proportional change in pressure resulting from variations in both gas quantity and container size.

Analyze the relationship for volume when pressure and temperature both double.

Volume is directly proportional to temperature and inversely proportional to pressure (

V = \frac{nRT}{P}

). Doubling both variables cancels out, leaving the volume unchanged.

To determine the net impact on volume when opposing proportional changes are applied to temperature and pressure.

Key Concept

Proportional scaling and algebraic manipulation of variables in scientific model equations

Estimated Time:1m 30s

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