Lesson 32: Combining the Gas Laws

(See pages 433-435)

In the last two lessons you learned how the volume of a gas is affected both by temperature and pressures which are applied to the gas. You learned that the relationship between temperature and the volume of a gas is a direct relationship (as the temperature goes up, so goes the volume and vice versa). Then you learned that the relationship between pressure and volume is an inverse relationship (as the pressure goes up, the volume goes down and vice versa). You also learned that in order to accurately measure a gas that it had to be placed under special conditions called STP. Measuring a gas at STP allows you to communicate with others the actual volume of gas present. In Lessons 30 and 31 you used mathematical relationships to convert gas volumes to STP. In this lesson we will combine those two steps into one.

Let's begin by looking at the mathematical relationship we used in Lesson 30 where we learned how temperature affects the volume of a gas. We used fractions like these:

T1 = VI

T2 / V2

Where T1 = the original temperature and T2 = the new temperature and where V1 = the original volume and V2 = the new volume.

With a little algebraic transformation (don't panic, just a little rearrangement of the variables), we can get this:

T1 V2 =T2 V1

Then, with solving for V2 (the new volume), we get:

V2 = VI (T2/T1)

This basically says that we can find the new volume by multiplying the original volume by the ratio of the new temperature to the old temperature.

If we take this expression and then add the relationship which tells how volume is affected with changes in pressure, we get this:

V2= VI (T2/ T1) (P1/P2) OR P1V1/T1 = P2V2/T2

This combines both of the relationships into one expression which basically says we can find the new volume by taking the original volume and multiplying by the ratio of new temperature to original temperature and by the ratio of original pressure to new pressure.

This may be getting confusing to you by this point so let's look at an example to help sort things out.

Example 1: Suppose you have collected 5.0 liters of oxygen gas at room conditions of 740 torr and 27 degrees C. What would the volume of oxygen be at STP?

To keep things organized, locate the following pieces of information:

T1 = 27 C

T2= 0 C

PI = 740 mm Hg

P2 = 760 mm Hg

V,= 5.0 L

V2 = this is what we're looking for, right?

Recall that before we can utilize the temperature measurements, we must first convert the Celsius values into Kelvin values. Do so by adding 273 to each value so our measurements now become:

T1 = 27 + 273 = _300_ degrees K.

T2 = 0 + 273 = __273 degrees K.

P1= 740 mm Hg

P2= 760 mm Hg

VI= 5L

V2 = ?

Before we put these values into our combined gas law formula, take a look at what happens when we convert the room conditions to STP. What happens to the temperature? Does it go up or down? ______Hopefully you said it goes down! If the temperature goes down, what happens to the volume of the gas? ______Yes, since this is a direct relationship, the volume should go down.

Now, consider the pressure side of the situation. The room pressure was 740torr. What happened when you go from room conditions to STP? Hopefully you said that the pressure goes up! Recall that the relationship between volume and pressure is an inverse relationship. As pressure goes up, volume comes down. So, again we should see a decrease in volume. Let's substitute in our values from our example and see the result!

Here's the combined gas law formula again:

V2 = VI (T2/ Ti) (P1/P2)

After substituting in our values, we get:

V2 = 5 L ( 273/300) (740/760)

Continuing to solve.....

V2 = 5 L (0.91) (0.97)

And after another step we get:

V2 = 5 L (88.27)

V2 = 4.41 L of oxygen gas

So, yes our volume did go down as expected (V1=5 L, V2 = 4.41 L)

Let's try another example.

Example 2: Tony was filling the tires on his four-wheeler with air before he went riding up into the mountains. At the gas station where he was filling the tires, the ambient temperature (ambient temperature means surrounding temperature or local conditions) was 34 C and the pressure was 762 torr. When he got up into the mountains the temperature was now 30 C and the pressure had fallen to 755 torn. If the volume of air inside each of Tony's tires was 24 L at the gas station, what was the new volume of the tires once he reached the top of the mountain?

Again, start by identifying measurements in the problem:

T1 = 34 + 273 = degrees K.

T2 = 30 + 273 = degrees K.

P1 = 762 mm Hg

P2= 755 mm Hg

V1 = 24 L

V2 = this is what we're looking for!

In looking at this situation, we see the temperature of the gas went ______which would make us think that the volume of the gas will also go______. However, take a look at the pressure change. The pressure goes ______and since pressure and volume have an inverse relationship, the volume should go ______. So, in this case we have two "competing" factors: one making the volume decrease and the other making the volume increase.

Let's substitute our values in the combined gas law formula and see what the final results will be:

V2 = V1 (T2/ T1) (P1/P2)

V2 = 24 L (303 / 307) (762 / 755)

V2 = 24 L ( 0.986) (1.01)

V2 = 24 L (0.996)

V2 = 23.9 L

So, in this case there was a slight decrease in overall volume of the air inside Tony's four wheeler tires.

Combining the Gas Laws

1.  Suppose you collected 10.0 L of hydrogen gas from a lab experiment when the room temperature was 34 degrees C and the atmospheric pressure read 762 torr. What would the volume of this hydrogen be at STP?

2.  Georgia had 3.5 L of argon gas at 33 degrees C and a pressure of 733 torr. If she heated the gas to 40 degrees C and decreased the pressure by 30 torr, what would the new volume of gas be?

3.  Sebastian combined zinc metal and hydrochloric acid to produce zinc chloride and hydrogen gas. If he collected 7.00 liters of hydrogen from this reaction when the room temperature was 24 C and the pressure was 755 torr, what would the volume be at STP?

4.  Tommy had 3.50 L of neon gas at 22 C. He heated it up to 32 C and increased the pressure to 780 torr. If the original pressure was 760 torr, what would the new volume of the neon be?

5.  Molly collected 10 liters of fluorine gas from a lab procedure. The room in which she collected it was at a temperature of 26 degrees C. If she placed it into a cooler where the temperature was 20 degrees C. what would the new volume of gas be. Pressures inside and outside the cooler were the same.