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Q. 1.7

Expert-verifiedFound in: Page 6

Book edition
1st

Author(s)
Daniel V. Schroeder

Pages
356 pages

ISBN
9780201380279

When the temperature of liquid mercury increases by one degree Celsius (or one kelvin), its volume increases by one part in 550,000 . The fractional increase in volume per unit change in temperature (when the pressure is held fixed) is called the thermal expansion coefficient, **β** :

$\beta \equiv \frac{\Delta V/V}{\Delta T}$(where V is volume, T is temperature, and Δ signifies a change, which in this case should really be infinitesimal if **β** is to be well defined). So for mercury, **β** =1 / 550,000 K^{-1}=1.81 x 10^{-4} K^{-1}. (The exact value varies with temperature, but between 0^{o}C and 200^{o}C the variation is less than 1 %.)(a) Get a mercury thermometer, estimate the size of the bulb at the bottom, and then estimate what the inside diameter of the tube has to be in order for the thermometer to work as required. Assume that the thermal expansion of the glass is negligible.(b) The thermal expansion coefficient of water varies significantly with temperature: It is 7.5 x 10 ^{-4} K^{-1} at 100^{o}C, but decreases as the temperature is lowered until it becomes zero at 4^{o}C. Below 4^{o}C it is slightly negative, reaching a value of -0.68 x 10^{-4}K^{-1} at 0^{o}C. (This behavior is related to the fact that ice is less dense than water.) With this behavior in mind, imagine the process of a lake freezing over, and discuss in some detail how this process would be different if the thermal expansion coefficient of water were always positive.

a) Diameter of the bulb is = 2 .8 x 10^{-3}m

b) Water has very different behavior of thermal expansion.

for mercury, **β** =1 / 550,000 K^{-1}=1.81 x 10^{-4} K^{-1}.

Lets assume that a typical mercury thermometer with cylindrical bulb having

height h= 1 cm = 1 x 10^{-2} mradius r=0.2 cm = 0.2 x 10^{-2} mand the scale on the thermometer is 1 mm (= 1 x 10^{-3} m) per degree.

Volume can be calculate by using

V = π r^{2}h

Substitute the values we get

$V=\pi {\left(0.2\times {10}^{-2}m\right)}^{2}\times \left(1\times {10}^{-2}m\right)\phantom{\rule{0ex}{0ex}}V=1.25\times {10}^{-7}{\mathrm{m}}^{3}$

Coefficient of thermal expansion is calculated as

$\beta =\frac{\Delta V}{V\Delta T}\phantom{\rule{0ex}{0ex}}Simplify,\phantom{\rule{0ex}{0ex}}\Delta V=\beta V\Delta T$

Substitute the value in above equation to calculate change in volume

$\Delta V=(1.81\times {10}^{-4}{K}^{-1})\times (1.256\times {10}^{-7}{m}^{3})\times (274.15K)\phantom{\rule{0ex}{0ex}}\Delta V=6.19\times {10}^{-9}{\mathrm{m}}^{3}$

The inner radius of the thermometer bulb r_{i} can be obtained from the change in volume

$\Delta V=\pi {{r}_{i}}^{2}\Delta l\phantom{\rule{0ex}{0ex}}Solveforradius\phantom{\rule{0ex}{0ex}}{r}_{i}=\sqrt{\frac{\Delta V}{\pi \Delta l}}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Substitutevalues\phantom{\rule{0ex}{0ex}}{r}_{i}=\sqrt{\frac{6.19\times {10}^{-9}{m}^{3}}{\pi \times {10}^{-3}m}}\phantom{\rule{0ex}{0ex}}{r}_{i}=1.4\times {10}^{-3}\mathrm{m}$

So diameter is 2 x radius = 2 x 1.4 x 10^{-3 }m = 2 .8 x 10^{-3}m

The thermal expansion coefficient of water varies significantly with temperature: It is 7.5 x 10 ^{-4} K^{-1} at 100^{o}C, but decreases as the temperature is lowered until it becomes zero at 4^{o}C. Below 4^{o}C it is slightly negative, reaching a value of -0.68 x 10^{-4}K^{-1} at 0^{o}C.

Thermal expansion coefficient of water behaves very differently.

β varies much in the liquid region.

As the temperature drops it decreases. But value of β is negative in between 0^{o}C and 4^{o}C i.e., ice generally seems to contract as temperature goes up to 4^{o}C.

If β were positive over the entire region of range , means always in one side of the curve.

This would lead to a water body (Lake, Pond) would start to freeze from bottom up Not from from the top down.

**Even at low density, real gases don’t quite obey the ideal gas law. A systematic way to account for deviations from ideal behavior is the virial**

**expansion,**

**$\mathit{P}\mathit{V}\mathbf{-}\mathit{n}\mathit{R}\mathit{T}\mathbf{(}\mathbf{1}\mathbf{+}\frac{\mathbf{B}\left(T\right)}{\left(V/n\right)}\mathbf{+}\frac{\mathbf{C}\left(T\right)}{{\left(V/n\right)}^{\mathbf{2}}}\mathbf{+}\mathbf{\cdots}\mathbf{)}$**

**where the functions $\mathit{B}\mathbf{\left(}\mathbf{T}\mathbf{\right)}$, $\mathit{C}\mathbf{\left(}\mathbf{T}\mathbf{\right)}$, and so on are called the virial coefficients. When the density of the gas is fairly low, so that the volume per mole is large, each term in the series is much smaller than the one before. In many situations, it’s sufficient to omit the third term and concentrate on the second, whose coefficient $\mathit{B}\mathbf{\left(}\mathbf{T}\mathbf{\right)}$ is called the second virial coefficient (the first coefficient is 1). Here are some measured values of the second virial coefficient for nitrogen (${\mathit{N}}_{\mathbf{2}}$):**

$\mathit{T}\mathbf{\left(}\mathbf{K}\mathbf{\right)}$ | $\mathit{B}\mathbf{(}\mathbf{c}{\mathbf{m}}^{\mathbf{3}}\mathbf{/}\mathbf{m}\mathbf{o}\mathbf{l}\mathbf{)}$ |

100 | –160 |

200 | –35 |

300 | –4.2 |

400 | 9.0 |

500 | 16.9 |

600 | 21.3 |

**For each temperature in the table, compute the second term in the virial equation, $\mathit{B}\mathbf{\left(}\mathbf{T}\mathbf{\right)}\mathbf{/}\mathbf{(}\mathbf{V}\mathbf{/}\mathbf{n}\mathbf{)}$, for nitrogen at atmospheric pressure. Discuss the validity of the ideal gas law under these conditions.****Think about the forces between molecules, and explain why we might expect $\mathit{B}\mathbf{\left(}\mathbf{T}\mathbf{\right)}$ to be negative at low temperatures but positive at high temperatures.****Any proposed relation between $\mathit{P}$, $\mathit{V}$, and $\mathit{T}$, like the ideal gas law or the virial equation, is called an equation of state. Another famous equation of state, which is qualitatively accurate even for dense fluids, is the van der Waals equation,$\mathbf{(}\mathbf{P}\mathbf{+}\frac{\mathbf{a}{\mathbf{n}}^{\mathbf{2}}}{{\mathbf{V}}^{\mathbf{2}}}\mathbf{)}\mathbf{(}\mathbf{V}\mathbf{-}\mathbf{n}\mathbf{b}\mathbf{)}\mathbf{=}\mathit{n}\mathit{R}\mathit{T}$where a and b are constants that depend on the type of gas. Calculate the second and third virial coefficients ($\mathit{B}$ and $\mathit{C}$) for a gas obeying the van der Waals equation, in terms of $\mathit{a}$ and $\mathit{b}$. (Hint: The binomial expansion says that ${\mathbf{(}\mathbf{1}\mathbf{+}\mathbf{x}\mathbf{)}}^{\mathbf{p}}\mathbf{\approx}\mathbf{1}\mathbf{+}\mathit{p}\mathit{x}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathit{p}\mathbf{(}\mathbf{p}\mathbf{-}\mathbf{1}\mathbf{)}{\mathit{x}}^{\mathbf{2}}$, provided that $\mathbf{\left|}\mathbf{p}\mathbf{x}\mathbf{\right|}\mathbf{\ll}\mathbf{1}$. Apply this approximation to the quantity ${\mathbf{[}\mathbf{1}\mathbf{-}\left(nb/V\right)\mathbf{]}}^{\mathbf{-}\mathbf{1}}$.)****Plot a graph of the van der Waals prediction for $\mathit{B}\mathbf{\left(}\mathbf{T}\mathbf{\right)}$, choosing $\mathit{a}$ and $\mathit{b}$ so as to approximately match the data given above for nitrogen. Discuss the accuracy of the van der Waals equation over this range of conditions. (The van der Waals equation is discussed much further in Section 5.3.)**

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