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I'm supposed to convert the heat capacity of water in $\frac{cal}{g^oC}$ to the heat capacity in $\frac{J}{(Kg)(K)}$

I was able to convert it from 1.00$\frac{cal}{g^oC}$ to 4184$\frac{J}{Kg^oC}$, but I don't know how to convert the Celcius part to Kelvins since $K=C+273$

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Since this is a measure of how much energy is required to change the temperature by a certain amount, you don't need to do any further conversion. It is the same since a change of $1 \text{ }^{\circ}\text{C}$ is the same as a change of $1 \text{ K}$. The offset of 273 doesn't matter.

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