Answer

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**Hint:**Bulk modulus is the measure of the compression of any substance. The bulk modulus is the ratio of infinite pressure increase to the volume of the substance. The reciprocal of the bulk modulus is known as compressibility.

**Formula used:**

The formula of the ideal gas equation is given by,

$ \Rightarrow PV = nRT$

Where pressure is P, the volume V, the number of moles is n, the universal gas constant is R and the temperature is T.

**Complete step by step solution:**

It is asked in the problem about the isothermal bulk modulus of an ideal gas at pressure ‘P’.

The formula of the ideal gas equation is given by,

$ \Rightarrow PV = nRT$

Where pressure is P, the volume V, the number of moles is n, the universal gas constant is R and the temperature is T.

Now differentiating the ideal gas equation we get,

$ \Rightarrow \left( {P\Delta V} \right) + \left( {V\Delta P} \right) = 0$

Here it is noted that the$\Delta T = 0$, as the temperature is constant.

$ \Rightarrow \dfrac{{\Delta V}}{V} = - \dfrac{{\Delta P}}{P}$

The isothermal bulk modulus is equal to,

$ \Rightarrow {B_{isothermal}} = - \dfrac{{\Delta P}}{{\left( {\dfrac{{\Delta V}}{V}} \right)}} = P$

So the isothermal bulk modulus with pressure P is equal to P.

**The correct answer for this problem is option A.**

**Additional information:**The bulk modulus is represented by the B or K. Mathematically bulk modulus is equal to $B = - V\dfrac{{dP}}{{dV}}$ where pressure is P and volume is V. The isothermal bulk modulus is the bulk modulus of the when temperature is constant.

**Note:**The isothermal bulk modulus is defined as the ratio of the change in the pressure to the fractional change in the volume at constant temperature. The derivation of the ideal gas gives the isothermal bulk modulus of the material.

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