\input{labpreamble}
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\subsubsection*{PHSX 222L\hfill Electrical Equivalent of Heat\hfill Name:\underline{\hspace*{2in}}}

\underline{Object:} The calorie is a common unit of thermal energy and
the energy of foods (though a food Calorie is 1000 physics calories)
and the joule is the common unit of chemical, mechanical, or
electrical energy.  Of course, because of the relationship between
energy and work, these units are also used for both energy and work.

In this experiment we intend to compare the calorie with the joule by 
comparing a measured decrease in the level of electrical energy (in joules) 
with the equivalent increase in thermal energy (in calories).

\underline{Theory:} When an electric current flows from a high
potential to a low potential through a circuit, the energy of the
flowing electrons is reduced because negative work is done on the
electrons.  But comparable positive work must be done by the electrons
on some other part of the system and hence the energy of that part of
the system increases.  The atoms do negative work on the electrons and
decrease their energy but the electrons do equal positive work on the
atoms and increase their energy.

In this experiment the electric current flows through a resistive
Nichrome wire (with resistance $R$).  The current interacts with the
metal atoms of the wire, does work on those atoms, and increases their
average energy and thereby heats it.  Thus, some of the energy of the
electrons now appears as thermal energy of the atoms.  The atoms of
the metal, which are in contact with the water, do work on the water
molecules and thereby increase their energy as well.
\begin{eqnarray}
\label{joule}({\rm decrease\ in\ electrical\ energy})&=&({\rm increase\ in\ thermal\ energy})\\
({\rm power})({\rm time})&=&({\rm specific\ heat})({\rm mass})({\rm 
temperature\ change})\\
({\rm current})({\rm voltage})({\rm time})&=&({\rm specific\ heat})({\rm mass})({\rm 
temperature\ change})\\
IVt & = & cm\Delta T,
\end{eqnarray}
 where $I$ is the current and $V$ the voltage, $c$ is the specific 
 heat, $m$ the mass, and $\Delta T$ the temperature change. Assuming $V=IR$,
the equation can be written two other ways:
\begin{equation}
I^{2}Rt=cm\Delta T
\end{equation}
\begin{equation}
\frac{V^{2}}{R}t=cm\Delta T
\end{equation}

The left hand side of this equation is in joules (the official
S.I. unit of energy), while the right hand side is in calories
($c_{\rm water} = 1.0$ cal/g \degree C); thus the number of joules in
one calorie can be determined.

A double-walled calorimeter is an excellent way to isolate the 
experiment from the surroundings.

\newpage

\underline{Procedure:} 
\vspace{-.25in}
\begin{enumerate}

\item Draw a diagram of the apparatus; include a wiring diagram.

\item Determine the resistance of the coil by measuring the current 
and voltage (check with an ohmmeter).

\item Fill the calorimeter can 3/4 full of cold water (about 
10$^{\circ}$C below room temperature).

\item Connect the voltmeter, ammeter, and heating coil into the 
circuit with the power supply.  (Do not turn on the power until 
checked by the instructor.)

\item Adjust the current to about 4.5 amps and begin to time the 
experiment.  Monitor the current closely throughout the experiment to 
keep it constant.  (Why is this necessary?)

\item Heat the water until it is as much above room temperature 
as it started below.

\item Record time, current, voltage, mass of water (take the water mass 
equivalent of the calorimeter, 14.7 g, into account), and temperature 
rise.

\item Divide the electrical energy consumed (in joules) by the 
thermal energy generated (in calories) to determine the number of 
joules in a calorie.

\item Compare with the accepted value.

\end{enumerate}


\underline{Questions:} 
A typical 50 gallon household hot water heater contains 189 kg of 
water.  Water from the water line is about 15$^{\circ}$C, and is 
heated by an electrical heating element having a resistance of 11 
$\Omega$.  The supply line voltage is 220 V.  (See similar problems 
in P27.39 and P27.48 in Serway.)
\vspace{-.2in}
\begin{enumerate}
\item How many calories of thermal energy must be delivered to the 
tank of water to heat it to bath temperature (40$^{\circ}$C)?

\item Calculate the power of the electric heating element.

\item Calculate the time required to heat up the water in the tank.

\item Calculate the cost of heating the water if the utility company 
charges 8 cents per kW-hr.

\item How much does your utility company charge per kW-hr?

\end{enumerate}


\underline{Conclusions:} 
\vspace{-.25in}
\begin{enumerate}
\item Discuss any sources of error that you feel might have affected 
your results.  Are some of these avoidable?  What effect would they 
have on your value?  Try to estimate the magnitude of the 
effects.

\end{enumerate}

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