The highest recorded waterfall in the world is found at Angel Falls in Venezuela. Its longest...

Question:

The highest recorded waterfall in the world is found at Angel Falls in Venezuela. Its longest single waterfall has a height of 807 m. If the water at the top of the falls is at 16.7 degree Celsius, what is the maximum temperature of the water at the bottom of the falls? Assume all the kinetic energy of the water as it reaches the bottom goes into raising the water's temperature.

Conservation of Energy

The energy changes its form during the different thermodynamic process but the sum of the energy of a system (which is isolated from the surroundings) remains constant. The increase in one type of energy causes a decrement in another type of energy.

Answer and Explanation:


Given Data

  • The height is: {eq}h = 807\;{\rm{m}} {/eq}
  • The initial temperature of the water is: {eq}{T_1} = 16.7^\circ {\rm{C}} {/eq}


The expression for the heat is,

{eq}Q = mc\left( {{T_2} - {T_1}} \right) {/eq}


Here, the specific heat of the water is {eq}c {/eq}, mass of the water is {eq}m {/eq} and the temperature of the water at the bottom is {eq}{T_2} {/eq}.


The expression for the kinetic energy is,

{eq}E = mgh {/eq}


From conservation of energy,

{eq}E = Q {/eq}


Substitute the known values,

{eq}\begin{align*} mgh &= mc\left( {{T_2} - {T_1}} \right)\\ gh &= c\left( {{T_2} - {T_1}} \right) \end{align*} {/eq}


Substitute the known values,

{eq}\begin{align*} \left( {9.81\;{\rm{m/}}{{\rm{s}}^2}} \right)\left( {807\;{\rm{m}}} \right) &= \left( {4183\;{\rm{J/kg}} \cdot {\rm{K}}} \right)\left( {{T_2} - 16.7^\circ {\rm{C}}} \right)\\ {T_2} - 16.7^\circ {\rm{C}} &= \dfrac{{\left( {9.81\;{\rm{m/}}{{\rm{s}}^2}} \right)\left( {807\;{\rm{m}}} \right)}}{{4183\;{\rm{J/kg}} \cdot ^\circ {\rm{C}}}}\\ {T_2} &= 1.892^\circ {\rm{C}} + 16.7^\circ {\rm{C}}\\ {T_2} &= 18.592^\circ {\rm{C}} \end{align*} {/eq}


Thus, the temperature of the water at the bottom is {eq}18.592^\circ {\rm{C}} {/eq}.


Learn more about this topic:

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Energy Conservation and Energy Efficiency: Examples and Differences

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