The volume $V$ of a sphere with radius $R$ is given by the formula: $$V = \frac{4}{3}\pi R^{3}$$
The coefficient of volume expansion $\gamma$ is related to the linear coefficient of thermal expansion $\alpha$ for isotropic solids by the relation: $$\gamma = 3\alpha$$
The change in volume $\Delta V$ for a temperature change $\Delta T$ is given by: $$\Delta V = V \gamma \Delta T$$
Substitute $V = \frac{4}{3}\pi R^{3}$ and $\gamma = 3\alpha$ into the equation: $$\Delta V = \left(\frac{4}{3}\pi R^{3}\right) (3\alpha) (\Delta T)$$ $$\Delta V = 4\pi R^{3}\alpha\Delta T$$
Final Answer: 4\pi R^{3}\alpha\Delta T
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