Question

The anomalous expansion characteristics of liquid water are crucial to many biological systems. Rather than an approximately constant value for the coefficient of volume expansion, the value for water changes drastically, as illustrated in the figure.

Below what temperature T does water shrink when heated?Express your answer numerically in degrees Celsius.
If the temperature of water at 30 ℃ is raised by 1 ℃, the water will expand. Atapproximately what initial temperature T will water expand by twiceas much when raised by 1 ℃?

Express your answer numerically in degrees Celsius.

Answer 1

Concepts and reason

The concept of coefficient of volume expansion is required to solve the problem.

First, use the expression of volume expansion and determine the coefficient of volume expansion that will result in a negative change in volume. Then from the graph, determine the temperature below which the change in the volume is negative.

After that determine the volume expansion when the temperature is raised by $1{{\rm{ }}^{\rm{o}}}{\rm{C}}$ from $30{{\rm{ }}^{\rm{o}}}{\rm{C}}$ . Then, determine the coefficient of volume expansion at which the volume expansion is twice as much when raised by $1{{\rm{ }}^{\rm{o}}}{\rm{C}}$ . Finally, use the graph and determine the temperature at which volume expansion is twice as much when raised by $1{{\rm{ }}^{\rm{o}}}{\rm{C}}$ .

Fundamentals

The change in volume of a substance when subject to a change in temperature depends on the amount of the temperature change, the coefficient of volume expansion, and initial volume. The change in the volume is given as,

$\Delta V = \beta {V_0}\Delta T$

Here, ${V_0}$ is the initial volume, $\beta$ is the coefficient of volume expansion, and $\Delta T$ is the change in temperature.

The change in the volume ( $\Delta V$ ) is the difference of the final volume (V) and initial volume ( ${V_0}$ ). That is,

$\Delta V = V - {V_0}$

(a)

Shrink in the volume of water on heating implies that the change in volume will be negative when the change in temperature is positive.

The change in the volume is given as,

$\Delta V = \beta {V_0}\Delta T$

Here, ${V_0}$ is the initial volume, $\beta$ is the coefficient of volume expansion, and $\Delta T$ is the change in temperature.

The change in the volume depends on the coefficient of volume expansion, initial volume, and change in temperature. When the change in temperature is positive, then the change in volume will be negative only if the coefficient of volume expansion is negative or less than zero.

From the given graph, the coefficient of volume expansion is negative below the temperature of ${4^{\rm{o}}}{\rm{ C}}$ . Hence, below ${4^{\rm{o}}}{\rm{ C}}$ temperature, water will shrink on heating.

(b)

The change in the volume is given as,

$\Delta V = \beta {V_0}\Delta T$

Here, ${V_0}$ is the initial volume, $\beta$ is the coefficient of volume expansion, and $\Delta T$ is the change in temperature.

Determine the change in volume when the temperature is raised by $1{{\rm{ }}^{\rm{o}}}{\rm{C}}$ from the initial temperature $30{{\rm{ }}^{\rm{o}}}{\rm{C}}$ .

Substitute $1{{\rm{ }}^{\rm{o}}}{\rm{C}}$ for $\Delta T$ in the above equation $\Delta V = \beta {V_0}\Delta T$ .

$\Delta V = \beta {V_0}\left( {1{{\rm{ }}^{\rm{o}}}{\rm{C}}} \right)$

The change in volume when the temperature is raised from the initial temperature T is, $\Delta V' = \beta '{V_0}\Delta T$

Substitute $2\Delta V$ for $\Delta V'$ and $1{{\rm{ }}^{\rm{o}}}{\rm{C}}$ for $\Delta T$ in the equation $\Delta V' = \beta '{V_0}\Delta T$ .

$2\Delta V = \beta '{V_0}\left( {1{{\rm{ }}^{\rm{o}}}{\rm{C}}} \right)$

Here, $\Delta V$ is the change in volume when temperature is raised by $1{{\rm{ }}^{\rm{o}}}{\rm{C}}$ from the initial temperature $30{{\rm{ }}^{\rm{o}}}{\rm{C}}$ .

Substitute $\beta {V_0}\left( {1{{\rm{ }}^{\rm{o}}}{\rm{C}}} \right)$ for $\Delta V$ in the above equation $2\Delta V = \beta '{V_0}\left( {1{{\rm{ }}^{\rm{o}}}{\rm{C}}} \right)$ and solve for $\beta '$ .

$\begin{array}{c}\\2\left( {\beta {V_0}\left( {1{{\rm{ }}^{\rm{o}}}{\rm{C}}} \right)} \right) = \beta '{V_0}\left( {1{{\rm{ }}^{\rm{o}}}{\rm{C}}} \right)\\\\\beta ' = 2\beta \\\end{array}$

Substitute $300 \times {10^{ - 6}}{{\rm{ }}^{\rm{o}}}{\rm{C}}$ for $\beta$ in the equation $\beta ' = 2\beta$ .

$\begin{array}{c}\\\beta ' = 2\left( {300 \times {{10}^{ - 6}}{{\rm{ }}^{\rm{o}}}{\rm{C}}} \right)\\\\ = 600 \times {10^{ - 6}}{{\rm{ }}^{\rm{o}}}{\rm{C}}\\\end{array}$

From the graph, the coefficient of volume expansion is $600 \times {10^{ - 6}}{{\rm{ }}^{\rm{o}}}{\rm{C}}$ when temperature is $70{{\rm{ }}^{\rm{o}}}{\rm{C}}$ .

Hence, at initial temperature of $70{{\rm{ }}^{\rm{o}}}{\rm{C}}$ , the water will expand by twice as much as it expands at $30{{\rm{ }}^{\rm{o}}}{\rm{C}}$ initial temperature.

Ans: Part a

Below ${4^{\rm{o}}}{\rm{ C}}$ temperature, the water will shrink on heating.

Part b

The initial temperature is $70{{\rm{ }}^{\rm{o}}}{\rm{C}}$ .

Answer 2

For water to shrink when heated,β has to be negative.

From the graph, water will shrink when heated below 4⁰C

For water to expand twice as much as it expands at 30⁰C,β has to be double.

β at 30⁰C = 300

β = 600 at 70⁰C

∴Water will expand twice as much as it expands at 30⁰C at 70⁰C.