Compute the bulk modulus of water from the following data:...

Compute the bulk modulus of water from the following data: Initial volume $=100.0$ litre, Pressure increase $=100.0 \mathrm{~atm}\left(1 \mathrm{~atm}=1.013 \times 10^{5} \mathrm{~Pa}\right)$, Final volume $=100.5$ litre. Compare the bulk modulus of water with that of air (at constant temperature). Explain in simple terms why the ratio is so large.

Key Concepts

Bulk Modulus

The bulk modulus is a measure of a material's resistance to compression. It is defined as the ratio of an applied pressure increase to the relative decrease in volume (fractional change in volume). In formulas, it is expressed as K = - V (?P/?V), where V is the original volume, ?P is the pressure change, and ?V is the change in volume. This concept applies universally to materials, whether liquids, solids, or gases, indicating how compressible (or incompressible) they are under pressure.

Volume Change and Pressure Variation

Understanding how a small change in volume relates to a change in pressure is essential for calculating the bulk modulus. The ability to connect these quantities allows one to quantify the stiffness of a fluid. In the context of the problem, converting units correctly for pressure and volume is key before applying the relationship, which underscores the importance of dimensional consistency and proper measurement conversion in physical calculations.

Incompressibility of Liquids

Liquids, such as water, are characterized by a very high bulk modulus, which means they undergo only a very slight change in volume even under significant pressure changes. This property reflects the dense packing of molecules in liquids. The high incompressibility of liquids is a fundamental reason why fluids like water can transmit pressure effectively, a concept that is central to understanding hydraulic systems and various geophysical phenomena.

Ideal Gas Behavior at Constant Temperature (Boyle's Law)

Gases behave differently compared to liquids because they are much more compressible. At constant temperature, gases obey Boyle's law, which states that the product of pressure and volume remains constant. This results in a much lower effective bulk modulus for gases, as even modest pressure increases can result in significant volume changes. This stark contrast with liquids highlights why the ratio of the bulk modulus of water to that of air is so large, emphasizing the fundamental difference in how densely packed the molecules are in these two states of matter.

Transcript

00:01 So this question belongs to the mechanical properties of solids in which we have given, we have to compute the bulk modulus of water that is beta w.

00:09 And we have given initial volume v initial it is equals to 100 .0 liter and change in pressure that is delta p it is equal to 100 .0 atm that is equals to 100 .0 multiplied by 1 .013 multiplied by 10 to the power 5 pascal and we have given the final volume v final this is equal to 100 .5 liter okay and we have to then compare this bulk modulus of water with the air so we know that bulk modulus beta it is equals to change in pressure delta p divided by change in volume divided by initial volume so we get from here delta p multiplied by initial volume divided by change in volume delta v and this change in volume will be we final minus v initial so we can write here v final minus v initial so substituting value so delta p it is 100 mudplared by 1 .013 muddclared by 10 to the power 5 pascal muddclared by v initial which is 100 liter devired by v final which is 100 liter minus v initial which is 100 liter okay so this quantity is in the liter and this is also in liter so we don't need to change it okay so from here we get bulk modulus b, this is equals to 2 .026 multiplied by 10 to the power 9 pascal.

01:31 So this become the bulk modulus of the water.

01:33 So this is bw actually.

01:35 So this become the answer for this problem.

01:38 Now, we will calculate the bulk modulus for the air.

01:41 So be air, it is equals to 1 .0 multiplied by 10 to the power 5 pascal.

01:47 So this value is smaller than this bulk modulus of water...

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