 Today I want to go over this gas law called the combined gas law and how we can use this law in different calculations as far as gases go. So first of all let's talk about what the combined gas law is just like it sounds. The combined gas law combines multiple laws that of Charles, Boyle's, and Galusex law to show us the relationship between pressure, volume, and temperature of a gas. And this is assuming that the amount of gas stays constant. So the formula for the combined gas law is P1 times V1 over T1 is equal to P2 times V2 divided by T2. These subscripts of 1 and 2 just refer to separate conditions that this gas is under. So the pressure of gas under condition 1 multiplied by the volume under condition 1 divided by the temperature is going to be equal to the pressure of the gas under condition 2 multiplied by the volume of gas under condition 2 divided by the temperature at condition 2. Once again like I said this assumes that your amount of gas does not change one way or the other. Now we can look at the combined gas law and if you happen to forget the other laws that Charles, Boyle's, and Galusex law we can also derive those laws from the combined gas law. We do that simply by eliminating the variable that's held constant. For example Boyle's law looks at the relationship between pressure and volume when we hold temperature constant. So if we get rid of these temperature values the only thing left is P1 times V1 is equal to P2 times V2 and this is the same equation used in Boyle's law. In terms of Charles law Charles law looks at the relationship between volume and temperature of a gas holding pressure constant. So if we get rid of the pressure variables we see that V1 over T1 is equal to V2 over T2. Once again this is the formula for Charles law. And then finally Galusex law looks at the relationship between pressure and temperature holding volume the same. If we get rid of volume then we find that Galusex law is equal to P1 over T1 equals P2 over T2. So we can derive each one of these three gas laws from the combined gas law. Now how do we use the combined gas law when given a problem which contains five of these six variables. I have an example over here. So our example question asks tells us that we have a gas with a volume of 675 milliliters. This gas is at 35 degrees Celsius and has a pressure of 0.850 atmospheres. If we change the temperature of this gas by decreasing it down to negative 95 degrees Celsius and the pressure increases to 1.06 atmospheres what is going to be the volume of this gas in condition 2. So in order to answer this problem let's first determine what the question gives us. What variables are we given? So the problem gives us under condition 1 it tells us that the pressure is 0.850 atmospheres. It also tells us under condition 1 this gas has a volume of 675 milliliters and it tells us that the temperature of the gas under condition 1 is 35 degrees Celsius. However we have a slight issue. We cannot use temperature in degrees Celsius when we're doing problems that refer to a gas. We must convert that to Kelvin first. In order to convert 35 degrees Celsius to Kelvin we simply add 273. When we do that we find that the temperature is 308 Kelvin. Now the problem also gives us some other variables. It tells us that we're changing the pressure of our gas from 0.850 atmospheres up to 1.06 atmospheres. So that's our pressure at condition 2. It also tells us what happens to temperature in condition 2 says the temperature decreases to negative 95 degrees Celsius. Once again our temperatures in Celsius it must be in Kelvin before we can use that value in a calculation. So in order to convert it to Kelvin we add 273 and find that the temperature is 178 Kelvin. So the only variable that we don't know is the one that we're going to solve for which is our volume under condition 2. Now we take all five of our variables and input them into the combined gas law. So P1 multiplied by V1 divided by temperature at condition 1 is going to be equal to our P2 multiplied by V2 which is what we don't know which is what we're going to solve for divided by T2 which is 178 Kelvin. In order to solve for V2 we first must move our variables on the right side of the equal sign over to the left side. In order to do that we simply multiply by the inverse. The inverse of 1.06 in the numerator is going to be 1.06 in the denominator. The inverse of 178 Kelvin on the denominator is 178 Kelvin in the numerator. When we do that to both sides on the right side of our equal sign what's going to happen is our variables will cancel out. Temperature and pressure will cancel out. So we are left with the one variable that we're trying to solve for. So what this tells us is V2 what we're trying to solve for is equal to 675 multiplied by 0.850 multiplied by 178 Kelvin and then we're going to divide that by 1.06 multiplied by 308 Kelvin. If we want to make sure that our equation is set up correctly what we can do is look at our units. We need to make sure that all of our units cancel out except for the one unit that we're solving for. So here Kelvin will cancel out with that Kelvin. Atmospheres will cancel out with atmospheres. The only unit we're left with is milliliters and what are we looking to solve for volume which is measured in milliliters. Whenever we put this equation and solve we find that V2 is equal to 313 milliliters. So our volume at condition 2 once the pressure and the temperature had changed we find is 313 milliliters and we solve that using this law called the combined gas law.