 ###### Jonathan Osbourne

PhD., University of Maryland
Published author

Jonathan is a published author and recently completed a book on physics and applied mathematics.

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# Bulk Modulus

Jonathan Osbourne ###### Jonathan Osbourne

PhD., University of Maryland
Published author

Jonathan is a published author and recently completed a book on physics and applied mathematics.

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Bulk modulus is a modulus associated with a volume strain, when a volume is compressed. The formula for bulk modulus is bulk modulus = - ( pressure applied / fractional change in volume). Bulk modulus is related to elastic modulus.

Okay let's stalk about bulk modulus, bulk modulus is another type of modulus that we measure for solid materials but we'll also measure this one for liquids and gases. Bulk modulus is associated with a volume strain, so what we're going to do is we're going to take our object and we're going to increase the pressure, we're going to push on it in all directions. All directions, and this is going to indicate a decrease it in its volume because we're pushing on it and it's going to get smaller. So as with all moduli we'll define the bulk modulus as the pressure that we applied divided by the fractional change in volume. Now this is a little bit of an issue, because we said if we applied pressure the volume is going to get smaller. So if the top is positive then the bottom is going to be negative and I don't want to define this thing as a negative number. So I'm going to give myself a minus sign.

We can do that because we're the ones that are defining the quantity in the first place. So we'll give ourselves this minus sign here and then we'll write down delta p over fractional change in volume, that's the change in volume over the volume giving ourselves a minus sign again. And then over here we'll write it in the way that you'll usually see it in the books. Now so sometimes this is called k instead of b but I like b so I'm going to use b. Alright so this bulk modulus is the pressure that's associated with a given decrease in volume. So the way that you'll usually see this quantity in a problem, in a Physics class, is you'll be asked to determine how much pressure is required to accomplish a 1% decrease in volume or a 5% decrease in volume. And all we'll need to do, to determine the answer to that is look over here at a table of bulk moduli. So we've got a table here, material and then the bulk modulus now as with all moduli, the bulk modulus is measured in Pascal's but it's a really, really, really, really big number.

I mean for steel it's going to take a lot of pressure for me to change the volume of steel. So for that reason I don't want to just write it in terms of Pascal's so I write it in terms of Giga Pascal's. So these numbers are given in terms of billions of Pascal's, billions of Newtons per square meter. Alright so let's say that I wanted to answer a question, how much pressure is required to produce a 1% decrease in volume of a sample of water? Alright so I got a 1% decrease in volume, now 1 nice way to think about the bulk modulus is that, that's the pressure required to give me 100% decrease in volume. Now that's actually an incorrect statement, because if I'm going to do 100% decrease in volume then the thing is not going to be there anymore. So the assumptions that I've made in defining this bulk modulus are actually not valid when the change in volume is that big.

But it's a useful way to think about it, because if I think about the bulk modulus, as the amount that I need for 100% decrease, then the amount that I need for 1% decrease is just going to be 1% of the bulk modulus. And that's actually the way that it goes, so if I want to know the pressure I'll say delta p equals 0.01, 1% times b and then I'll jut write it out. So it'll be 10 to the minus 2 times and the bulk modulus for water is 2.2 Giga Pascal's so it'll be 2.2 times 10 to the nine Pascal's. And then I'll just multiply and I'll get 2.2 times 10 to the seven Pascal's right? Or 22 million Pascal's, and so that's the answer to that problem.

Now just thinking about bulk modulus somebody gives you the bulk modulus of a material, essentially what they're doing is they're telling you how hard it is to change its volume. How hard it is to compress it, the bigger the bulk modulus, the more difficult it is to squeeze those molecules together. So we can see from this table that diamond is very, very difficult to squeeze together. And so it has a very large bulk modulus, alright so that's bulk modulus enjoy.