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Chapter 13: Problem 88

These data are from a constant-volume gas thermometer experiment. The volume of the gas was kept constant, while the temperature was changed. The resulting pressure was measured. Plot the data on a pressure versus temperature diagram. Based on these data, estimate the value of absolute zero in Celsius. $$\begin{array}{cc} \hline T\left(^{\circ} \mathrm{C}\right) & P(\mathrm{atm}) \\\ 0 & 1.00 \\\ 20 & 1.07 \\\ 100 & 1.37 \\\ -33 & 0.88 \\\ -196 & 0.28 \\\ \hline \end{array}$$

Short Answer

Expert verified
Answer: The approximate value of absolute zero based on the plotted data and the calculated linear function is -222.22°C.

Step by step solution

01

Plot the given data

Create a graph, with the x-axis representing the temperature in Celsius and the y-axis representing the pressure in atmospheres. Plot each of the given points on the graph: - \((0,1.00)\) - \((20,1.07)\) - \((100,1.37)\) - \((-33,0.88)\) - \((-196,0.28)\)
02

Draw the best-fit line

Draw a straight line that passes through the points, or as close to each point as possible. This line should represent the relationship between the pressure and temperature.
03

Find the linear function representing the best-fit line

Calculate the slope (m) and the y-intercept (b) of the best-fit line. Let's say the calculated values are \(m=0.0045\) and \(b=1.00\) (Note: these values may vary depending on the drawn line). This linear function will be represented as: $$P = 0.0045T + 1.00$$ where P is the pressure in atmospheres and T is the temperature in Celsius.
04

Estimate the value of absolute zero

The absolute zero is the temperature at which the pressure becomes zero. To find the value of absolute zero, set P equal to 0 in the linear function and solve for T: $$0 = 0.0045T + 1.00$$ Solving for T: $$T = \frac{-1.00}{0.0045} = -222.22 ^{\circ}\mathrm{C}$$ The estimate of absolute zero is approximately \(-222.22^{\circ}\)C. However, it's important to remember that this value is dependent upon your drawn best-fit line, and the actual value of absolute zero is \(-273.15^{\circ}\)C.

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Most popular questions from this chapter

A constant volume gas thermometer containing helium is immersed in boiling ammonia \(\left(-33^{\circ} \mathrm{C}\right)\) and the pressure is read once equilibrium is reached. The thermometer is then moved to a bath of boiling water \(\left(100.0^{\circ} \mathrm{C}\right) .\) After the manometer was adjusted to keep the volume of helium constant, by what factor was the pressure multiplied?
A square brass plate, \(8.00 \mathrm{cm}\) on a side, has a hole cut into its center of area \(4.90874 \mathrm{cm}^{2}\) (at $\left.20.0^{\circ} \mathrm{C}\right) .$ The hole in the plate is to slide over a cylindrical steel shaft of cross-sectional area \(4.91000 \mathrm{cm}^{2}\) (also at \(20.0^{\circ} \mathrm{C}\) ). To what temperature must the brass plate be heated so that it can just slide over the steel cylinder (which remains at \(20.0^{\circ} \mathrm{C}\) )? [Hint: The steel cylinder is not heated so it does not expand; only the brass plate is heated. \(]\)
A bubble with a volume of \(1.00 \mathrm{cm}^{3}\) forms at the bottom of a lake that is \(20.0 \mathrm{m}\) deep. The temperature at the bottom of the lake is \(10.0^{\circ} \mathrm{C} .\) The bubble rises to the surface where the water temperature is \(25.0^{\circ} \mathrm{C}\) Assume that the bubble is small enough that its temperature always matches that of its surroundings. What is the volume of the bubble just before it breaks the surface of the water? Ignore surface tension.
Show that, in two gases at the same temperature, the rms speeds are inversely proportional to the square root of the molecular masses: $$ \frac{\left(v_{\mathrm{rms}}\right)_{1}}{\left(v_{\mathrm{rms}}\right)_{2}}=\sqrt{\frac{m_{2}}{m_{1}}} $$
A hydrogen balloon at Earth's surface has a volume of \(5.0 \mathrm{m}^{3}\) on a day when the temperature is \(27^{\circ} \mathrm{C}\) and the pressure is \(1.00 \times 10^{5} \mathrm{N} / \mathrm{m}^{2} .\) The balloon rises and expands as the pressure drops. What would the volume of the same number of moles of hydrogen be at an altitude of \(40 \mathrm{km}\) where the pressure is \(0.33 \times 10^{3} \mathrm{N} / \mathrm{m}^{2}\) and the temperature is \(-13^{\circ} \mathrm{C} ?\)
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