At a temperature t^∘ C, a liquid is completely filled in a...

At a temperature $t^{\circ} \mathrm{C}$, a liquid is completely filled in a spherical shell of copper. If $\Delta T$ increases temperature of the liquid and the shell, then the outward pressure $d P$ on the shell resulted from increase in temperature is given by: (Given, $K=$ Bulk modulus of the liquids, $\gamma=$ coefficient of volume expansion, $\alpha=$ coefficient of linear expansion of the material of the shell) (a) $\frac{K}{2}(\gamma-3 \alpha) \Delta T$ (b) $K(3 \alpha-\gamma) \Delta T$ (c) $3 \alpha(K-\gamma) \Delta T$ (d) $\gamma(3 \alpha-K) \Delta T$

At a temperature $t^{\circ} \mathrm{C}$, a liquid is completely filled in a spherical shell of copper. If $\Delta T$ increases temperature of the liquid and the shell, then the outward pressure $d P$ on the shell resulted from increase in temperature is given by: (Given, $K=$ Bulk modulus of the liquids, $\gamma=$ coefficient of volume expansion, $\alpha=$ coefficient of linear expansion of the material of the shell)
(a) $\frac{K}{2}(\gamma-3 \alpha) \Delta T$
(b) $K(3 \alpha-\gamma) \Delta T$
(c) $3 \alpha(K-\gamma) \Delta T$
(d) $\gamma(3 \alpha-K) \Delta T$

Key Concepts

Bulk Modulus

Bulk modulus is a measure of a material's resistance to uniform compression. It is defined as the ratio of an infinitesimal pressure increase to the resulting relative decrease in volume. In the context of liquids, a high bulk modulus indicates that the liquid is relatively incompressible, which is critical for understanding how a temperature-induced volume change can result in significant pressure changes if the liquid is confined.

Coefficient of Volume Expansion

The coefficient of volume expansion quantifies how much a material's volume changes per unit temperature increase. For a liquid, this parameter determines the extent of its volumetric expansion with rising temperature. When the liquid is confined in a container, such as a spherical shell, this expansion can lead to an increase in internal pressure if the container does not expand equivalently.

Coefficient of Linear Expansion

The coefficient of linear expansion describes the rate at which a material's dimensions (length, and in isotropic materials, all dimensions) change with temperature. For solids like copper, this coefficient is used to determine how much the container expands when the temperature increases. The mismatch between the expansion of the container and the volume expansion of the confined liquid results in the buildup of pressure.

Thermal Stress Due to Constrained Expansion

Thermal stress arises when different materials or components expand at different rates due to temperature changes, leading to internal forces being generated in the system. In a scenario where a liquid expands inside a container, if the container's expansion is insufficient to accommodate the increased volume, the liquid's expansion is constrained, generating an outward pressure on the container walls. The analysis of these stresses requires an interplay between the liquid's bulk modulus and the expansion coefficients of both the liquid and the container material.

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