 One small thing that was ignored earlier is change in density with the temperature. Suppose you have let us say liquid of volume V, you increase the temperature, will it mass change, will it volume change, so will it density change, extra change, so how density will vary, that is what the next topic is, so I come variation of, so density is nothing but mass divided by volume, original density let us say is rho naught, which is m naught by V naught. Now, if you increase the temperature, the density should be equal to mass remains n, m naught and volume becomes V naught 1 plus gamma delta t, this is equal to m naught by V naught times 1 plus gamma delta t raise to power minus 1, do you remember binomial approximation, do you remember binomial approximation goes like this, 1 plus x raise to power y is approximately equal to 1 plus x y, if mod of x is very less than 1, as in it is like 0.0001, so similarly here if you use binomial expansion, binomial approximation and naught by V naught is rho naught, so this is rho naught 1 minus gamma delta t, same volumetric expansion coefficient is coming, but density is original density into 1 minus gamma delta t, see here y is minus mass, here y can be anything, no no y can be anything, y can be negative also, you do not know, if y is negative it will be negative, head outs here, no let us quickly do a numerical on this, then we will proceed to next property, a solid floats in a liquid at 20 degree Celsius, with 75 percent of it immerse in the liquid, 75 percent is inside, when the liquid is heated to 100 degree Celsius, liquid's temperature goes to 100 degree Celsius, when this happens, the same solid floats with 80 percent of it immerse in the liquid, this happens then 80 percent is inside the liquid, you need to find coefficient of expansion for the liquid, assume that volume of solid remains constant between 20 and 80, 20 and 100 degree Celsius, volume will, solid will not expand, you will know that, fine, you need to find gamma for liquid, how much, did you get, see what will happen is that, what there is a weight of the liquid, that should be equal to weight of the object only, because buoyant force is balancing the volume, so if this is the solid, if total volume is V, then 3 by 4 times V is inside the liquid, so buoyant force will be what, density of liquid into volume is placed, which is 3 by 4 V equals g, this is the buoyant force, any doubt here, this buoyant force is acting upwards, and then there will be a gravity force, let us say which is mg, acting downwards, so initially rho times, this is initial density of the liquid, rho naught 3 by 4 V into g, this is equal to mg, any doubt here, no, this is your first equation, now after you heat it, after you heat it, density of liquid will change, fine, let us say new density is rho, so rho into, now 80 percent is inside the liquid, so 80 by 100, that is 8 by 10 times V into g, this should be equal to mg, any doubts, anything, now this V remains V only, repeating it, this volume is what, volume of the solid, which does not expand, we are ignoring the expansion of solid, so it remains V, so just divide these two equations, you will get rho naught into 3 by 4 divided by rho into 4 by 5 is equal to 1, fine, so from here you will get rho is equal to or rho into 4 by 5 will be equal to rho naught into 3 by 4, okay, so rho will be equal to rho naught into 4 by 5 is equal to 15 by 16, okay, any doubts, nothing, so how you get gamma from here, this should be equal to what, rho naught, what is delta T, 80 by 100 from 28, this is 80, rho naught and rho naught will be answered, so 80 times gamma is equal to 1 minus 15 by 16, so gamma is 1 by 80 times 1 by 16, did you get this kind of, did you follow this concept, like this, right, so this is how you have to look this particular, this is what, 18 to 16, 1180, which is 8.47 into 10 to the power minus 4, 1 divided by this, fine, so the next property we will be going to talk about is heat capacity, heat capacity as the name suggests, it tries to quantify how much heat an object can store, it is a capacity of storing heat by an object, okay, and one of you said there, you can say there, no, no, it was all the way here, okay, so it should quantify how much heat an object can store, okay, like for example if I give you a bucket, if I give you a bucket like this, it is a huge bucket, you cannot measure its dimensions for example, okay, and you need to estimate what is the, let us say capacity of this bucket, compared to this bucket, okay, all you have is 1 litre water, then what you should do, pour it, that is all you can do right now, pour it and you will see that the height is more here and less here, fine, so with respect to how much water level is, you can estimate the capacity, you can say that capacity of this bucket is more, fine, so it is like whatever amount of liquid you are giving inside the bucket, divided by whatever is the height, this height should give you an indication of what is the capacity of the bucket, lesser the height, more is the capacity, fine, so I am just giving you a very crude example of how you should understand the heat capacity, fine, so just like capacity to store water, similarly we are talking about capacity to store heat, when you give heat to an object, what will increase, temperature will increase, height will not increase, height is for water, to give heat to the substance is temperature will increase, so similarly if I take a measure of how much temperature has increased for a given amount of heat and giving it, I can call this as heat capacity, okay, this is heat capacity which can be generated by let us see, how many objects you have in this room, more than 100, if you just open that CPU, more than 100 will come out from that CPU itself, okay, similarly you will see more than 100 objects in this classroom itself, imagine how many objects are there in the world, you cannot count, they are countless and then you cannot use this term heat capacity and go on listing down all the heat capacities of each and every object in the world, it become very tedious process, okay, but then if you know heat capacity, if you know C for an object, if you know heat capacity for an object, how much heat it can absorb for let us say delta T rise, C into delta T, simply, right, so formula wise it is very simple, but practically listing down all the heat capacities for each and every object in this world is very difficult, okay, so that is the reason why there has to be a simpler way to define heat capacity, so that I take into account of all the material in the earth, okay, now ultimately any material you take it is made up of elements, basic elements, okay, how many elements you have in that periodic table, 108 or 109 like that, okay, so can I safely say that entire earth is made up of those 108 or 109 elements, right, and out of those 108 and 109 there are just 6 or 7 of them which constitute more than 90 percent, fine, so if I define some capacity related thing for only those material which are like purest form, then all the objects are made up of those material only, getting it, for example this chair, it is made up of let us say wood, then iron, then cushion, cloth and everything is there, so if I know separately that for iron this is the capacity for wood, this is the capacity for cloth, for this cloth, this is, then I can you know individually cater to, like for example for iron I can find out how much heat it can absorb, for wood I can take it separately, right, once I split it, whatever object it is, if I split into its basic constituents then it become very easy to handle, okay, but problem lies here, that for example if I take 1 kg of iron, okay, if it can absorb let us say 10 joules of heat and it can create only 1 degree Celsius rise, okay, now if I take 2 kg of iron then if it absorb 10 joules of heat, how much rising temperature will be approximately half, half mass is double, we are getting it, so even though you are considering the material in its purest form but then also heat capacity is different for different masses fine, so it is a good idea to define heat capacity as per unit mass, per kg what is the heat capacity, fine, you understand, so that is the reason why there is this term just called specific heat capacity which is heat capacity per kg, okay, so write down specific heat capacity