 So let's go through this problem which is asking us to calculate the mole fraction, the molarity and the molality of a 12.2% by mass solution of HNO3 in water and we are given the density of the solution which is equal to 1.038 grams per centimeter cube. So you can pause the video here and give this a try and then we'll continue. Okay so there are different ways of solving this problem but usually in the cases where a mass percent is given it becomes very easy to solve these type of problems if we make a simple assumption. So since we are given that the mass percent is 12.2 let's say that we assume that we have taken 100 grams of the solution. Now if we have taken 100 grams of the solution and it is a 12.2% solution by mass that means 12.2 grams is going to be the solute that is HNO3 and the mass of the solvent is going to be 100 minus the mass of the solute. So if we calculate we find that the mass of HNO3 is 12.2 grams and for this solution the mass of the solvent is 87.8 grams and why we are doing this is because now for this solution which is 12.2 mass percent we have the mass of the solute and the mass of the solvent and we know the molar masses of HNO3 and water. So since we have the masses and we have their molar masses we can calculate the number of moles of HNO3 and the number of moles of water and from that we can see that we can calculate the mole fraction easily. So let's see how we can do that. So to calculate the number of moles we can use this formula. We know that the number of moles is equal to the mass divided by the molar mass. So in case of HNO3 we took the mass to be 12.2 grams and we were given that the molar mass was 63 grams per mole. So if you calculate the number of moles of HNO3 it comes out to be 0.1937 and similarly in the case of water the mass of the water was 87.8 grams and we know that the molar mass of water is 18 grams per mole. So if you calculate this the number of moles of water comes out to be 4.8778. Now since we know the number of moles of HNO3 and water we can find the total number of moles which will be equal to 5.0715. Now we were asked to calculate the mole fraction of HNO3 and we know that the mole fraction of HNO3 will be equal to the number of moles of HNO3 divided by the total number of moles and we just calculated these values. So if we plug in these values the mole fraction of HNO3 will be equal to 0.1937 that is the number of moles of HNO3 divided by the total number of moles that is 5.0715 and if we solve this we find the mole fraction of HNO3 to be equal to 0.038 and if we wanted to find the mole fraction of water it will be equal to 1 minus this value because sum of the mole fraction of HNO3 and the mole fraction of water will be equal to 1. Now next step is to calculate the molarity. To calculate the molarity let's start with the definition that is the number of moles divided by the volume of solution in liters. Now we calculated the number of moles in the previous step. To calculate the volume of the solution we were given the density of the solution and so we can use the formula for the density which is equal to the mass of the solution divided by the volume of the solution and since we assume the mass of the solution to be 100 grams and we know the value of the density we can calculate the volume by plugging in the known values. So if we rearrange this we can calculate the volume in centimeter cube as equal to 96.339 centimeter cube but for the calculation of molarity we need a volume in liters and we know that 1cc is equal to 1ml and so the volume in liters will be 96.339 into 10 to the power minus 3 liters. So now we know the number of moles from before and we have also calculated the volume of solution in liters. So if we plug in these values we can calculate the molarity by solving this that is the number of moles divided by the volume of the solution in liters and this number comes out to be 2.011 moles per liter which is the molarity. Now next step let's calculate the molality and lastly to calculate molality let's use the definition that is the number of moles of solute divided by the mass of the solvent in kg. So we had already calculated the number of moles of the solute and for the mass of the solvent because we had assumed a 100 gram solution we calculated the mass of the solvent to be 87.8 grams so if we plug in these values we get 0.1937 which was the number of moles of the solute that we calculated before divided by the mass of the solvent in kgs which is why there is this factor of 10 to the power minus 3 and if we calculate this value the molality comes out to be 2.206 moles per kg.