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1

I think you are confused between potential difference and energy. The energy used up can vary, if the current flowing through the resistor varies. First of all, a battery is 'not' a constant energy source. It's a constant potential source. Secondly, if more current flows, more energy is dissipated, even though the per capita energy is constant. Now, to ...

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Happy birthday. Once upon a time, there was a king and he had a sister who also liked coffee... Or let me omit this portion of the answer. A battery is motivating the electrons to buy a round trip ticket around the circuit by the voltage $V$. The voltage is nothing else than the energy $E$ per unit charge $Q$, i.e. $V=E/Q$; the unit 1 volt is nothing else ...

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Consider for a moment, a cell that is not connected to a circuit, i.e., there is no path for current external to the cell. The chemical reactions inside the cell remove electrons from the cathode and add electrons to the anode. Thus, as the chemical reactions proceed, an electric field builds between the anode and cathode due to the differing charge ...

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This quote is only for free Bose gas. For interacting Bose gas, of course the chemical potential can be above the single particle ground state. This is common in the cold atom Bose-Einstein condensates.

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