While maintaining a constant volume, a steel ball initially at a pressure of 1.0 10⁵ Pa is heated from 20^ C to 120^ C . Determine the pressure inside the ball. The coefficient of linear expansion of steel is 12 10⁻⁶ ^ C⁻¹ and the bulk modulus of steel is 1.6 10¹¹ N m ⁻² .

Step-by-step solution

The fractional change in volume due to a temperature change T is given by V V = T = 3 T To maintain a constant volume, the increase in pressure must counteract this thermal expansion. The required change in pressure P is related to the bulk modulus B by P = B V V = B (3 ) T Given values are: B = 1.6 10¹¹ N m ⁻² = 12 10⁻⁶ ^ C ⁻¹ T = 120^ C - 20^ C = 100^ C Substituting these values: P = (1.6 10¹¹) (3 12 10⁻⁶) 100 P = 1.6 10¹¹ 36 10⁻⁴ P = 57.6 10⁷ Pa = 5.76 10⁸ Pa The final pressure inside the ball is the sum of the