How do I calculate the heat lost or gained by surroundings (Q surr) given mass ($m$), change in temp ($\Delta T$), and specific heat ($c$)? What equation would I use?
How can I tell whether it's lost or gained?
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That depends on what you mean by "heat". If you mean energy then you can calculate the energy required to cause the temperature change using the following equation: Energy = mass * specific heat capacity * temperature change (Q = mcθ).