In Fig. 26-37, a resistance coil, wired to an external...

In Fig. $26-37,$ a resistance coil, wired to an external battery, is placed inside a thermally insulated cylinder fitted with a frictionless piston and containing an ideal gas. A current $i=240 \mathrm{mA}$ flows through the coil, which has a resistanc $R=550 \Omega$ . At what speed $v$ must the pis ton, of mass $m=12$ kg, move upward ir order that the temperature of the gar remains unchanged?

In Fig. $26-37,$ a resistance coil, wired to an external battery, is placed inside a
thermally insulated cylinder fitted with a
frictionless piston and containing an
ideal gas. A current $i=240 \mathrm{mA}$ flows through the coil, which has a resistanc
$R=550 \Omega$ . At what speed $v$ must the pis
ton, of mass $m=12$ kg, move upward ir
order that the temperature of the gar
remains unchanged?

Key Concepts

Joule Heating

Joule heating refers to the process by which electrical energy is converted into thermal energy in a resistor. The power generated as heat is given by the formula P = i²R, and this concept is fundamental in understanding how electrical circuits interface with thermodynamic systems by providing a quantifiable amount of energy input.

First Law of Thermodynamics in Isothermal Processes

For an ideal gas undergoing an isothermal (constant temperature) process, the first law of thermodynamics mandates that any energy added to the system as heat must be exactly offset by the work done by the gas, so that the internal energy remains unchanged. This energy balance is crucial when considering processes where temperature is held constant despite energy conversions.

Mechanical Work and Energy Transfer

Mechanical work in this context is the work done by the gas as it expands and moves the piston. The work done per unit time (power) by the gas can be related to the force (which, in scenarios involving a weight on a piston, corresponds to the gravitational force) and the velocity of the piston. This concept ties together the mechanical energy output and the energy input from Joule heating.

Energy Balance in Thermodynamic Systems

The principle of energy balance in thermodynamic systems requires that all energy entering the system as heat (in this case, from electrical resistance heating) is accounted for by an equivalent amount of work done by the system. This balance ensures that, for an isothermal process, the system’s temperature remains unchanged as the energy input is fully utilized in doing work.

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