Heat

Step 1: First Law of Thermodynamics.
The first law of thermodynamics states: where is the heat added to the system, is the change in internal energy, and is the work done. For an isothermal process, the temperature is constant, so the change in internal energy . Thus, the heat added is equal to the work done:
Step 2: Work Done in an Isothermal Process.
The work done during an infinitesimal expansion or compression of a gas is given by: where is the pressure and is the infinitesimal change in volume. Using the ideal gas law , we can substitute into the equation for work:
Step 3: Integrating the Work.
To find the total work done during the isothermal process, we integrate from the initial volume to the final volume : Since is constant, we can take it outside the integral: The integral of is , so:

Step 1: Newton's Law of Cooling.
According to Newton's law of cooling, the rate of heat loss is given by: where is the heat transfer constant, is the temperature of the object, and is the surrounding temperature.
Step 2: Calculate Rate of Heat Loss.
For the metal sphere at 827°C: For the metal sphere at 427°C: Thus, the rate of heat loss is twice as much at 827°C compared to 427°C.