The ideal gas law, expressed as \( PV = nRT \), serves as a foundational equation in understanding the relationships between pressure (P), volume (V), number of moles (n), and temperature (T) of an ideal gas. When faced with problems involving two sets of values for these variables, we can rearrange the ideal gas law to derive new equations that facilitate calculations.In scenarios where you have two pressures and two temperatures, or two volumes with two moles, the ideal gas law can be adapted to compare the initial and final states of a gas. This is particularly useful in applications such as gas expansion or compression, where you might need to find an unknown variable while knowing the others.For example, if you have an initial state represented as \( P_1, V_1, n_1, T_1 \) and a final state as \( P_2, V_2, n_2, T_2 \), you can set up the relationship as follows:\[\frac{P_1 V_1}{n_1 T_1} = \frac{P_2 V_2}{n_2 T_2}\]This equation allows you to solve for any unknown variable, provided you have the necessary information about the other variables. By understanding how to manipulate the ideal gas law in this way, you can effectively tackle a variety of problems involving gases under different conditions.
11 Gases
The Ideal Gas Law Derivations
11 Gases
The Ideal Gas Law Derivations: Videos & Practice Problems
Topic summary
The Ideal Gas Law Derivations come from rearranging the ideal gas law, \(PV=nRT\) , when a problem involves two values for selected variables. The key idea is to identify which quantities change and which remain constant, then remove the constants to create a simpler relationship between the changing terms.
This leads to common derived forms such as \(P_1V_1=P_2V_2\) , \( \frac{V_1}{n_1}=\frac{V_2}{n_2}\) , and \( \frac{P_1}{T_1}=\frac{P_2}{T_2}\) . In these derivations, pressure, volume, moles, and temperature must be tracked carefully, and temperature must be converted to Kelvin before calculation.
The Ideal Gas Law Derivations are a convenient way to solve gas calculations involving 2 sets of the same variables.
Ideal Gas Law Derivations
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Concept
The Ideal Gas Law Derivations
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Problem
A sample of nitrogen dioxide gas at 130 ºC and 315 torr occupies a volume of 500 mL. What will the gas pressure be if the volume is reduced to 320 mL at 130 ºC?
A
0.65 atm
B
0.83 atm
C
0.92 atm
D
1.6 atm
E
2.7 atm
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Problem
A cylinder with a movable piston contains 0.615 moles of gas and has a volume of 295 mL. What will its volume be if 0.103 moles of gas escaped?
A
0.176 L
B
0.217 L
C
0.246 L
D
0.361 L
E
1.28 L
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Problem
On most spray cans it is advised to never expose them to fire. A spray can is used until all that remains is the propellant gas, which has a pressure of 1350 torr at 25 ºC. If the can is then thrown into a fire at 455 ºC, what will be the pressure (in torr) in the can?
a) 750 torr
b) 1800 torr
c) 2190 torr
d) 2850 torr
e) 3300 torr
A
750 torr
B
1800 torr
C
2190 torr
D
2850 torr
E
3300 torr
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The combined gas law is derived by starting with the ideal gas law, . When dealing with two different states of a gas, where the amount of gas (n) and the gas constant (R) remain constant, you can set up the equation for both states: . This equation relates the initial and final states of pressure, volume, and temperature, allowing you to solve for any unknown variable when two sets of conditions are given.
Rearranging the ideal gas law is crucial because it allows you to adapt the equation to different problem scenarios involving changes in pressure, volume, temperature, or moles of gas. When a gas undergoes a change from one state to another, you often have two sets of conditions. By rearranging the ideal gas law, you can derive formulas that relate these two states, such as the combined gas law or other specific forms. This flexibility helps in solving problems where some variables change while others remain constant, making it easier to calculate unknown values and understand gas behavior under varying conditions.
To calculate the number of moles when pressure, volume, and temperature change, you start with the ideal gas law: . If you have two sets of conditions, you can write the equation for each state. If the gas amount changes, you can solve for the new number of moles by rearranging the equation: . By plugging in the new pressure, volume, and temperature values, you can find the new moles of gas. This is especially useful in reactions or processes where gas quantity changes.
Common mistakes include not converting all units to the correct SI units (pressure in atm or Pa, volume in liters or cubic meters, temperature in Kelvin), mixing up initial and final states, and forgetting to keep the gas constant (R) consistent. Another error is neglecting to convert temperature to Kelvin, which is essential because the ideal gas law requires absolute temperature. Also, students sometimes fail to recognize when the number of moles changes and incorrectly apply the combined gas law without accounting for this. Careful attention to these details ensures accurate use of the ideal gas law derivations.
According to the ideal gas law, temperature changes directly affect pressure and volume. If the amount of gas and one variable (pressure or volume) are held constant, increasing temperature will increase the other variable. For example, at constant volume, increasing temperature increases pressure, as shown by . At constant pressure, increasing temperature increases volume, expressed as . These relationships are derived from the ideal gas law by rearranging it to compare two states, helping predict how gases respond to temperature changes.
