A fluid undergoes a thermodynamic process such that its pressure P and volume V vary with a parameter t as P = a t^3 and V = b t^2 , where a and b are positive constants. The bulk modulus of the fluid in terms of its pressure P is

Step-by-step solution

The bulk modulus is defined as B = -V dP dV . Given the parametric equations: P = a t^3 V = b t^2 Differentiate both with respect to t : dP dt = 3at^2 dV dt = - 2b t^3 Now, find dP dV using the chain rule: dP dV = dP dt dV dt = 3at^2 - 2b t^3 = - 3a 2b t^5 Substitute this into the bulk modulus formula: B = -V dP dV = - ( b t^2 ) (- 3a 2b t^5 ) B = 3a 2 t^3 Since P = a t^3 , we can substitute P back into the expression: B = 3 2 P Answer: 3P 2