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Chapter 13: Problem 52

In intergalactic space, there is an average of about one hydrogen atom per \(\mathrm{cm}^{3}\) and the temperature is \(3 \mathrm{K}\) What is the absolute pressure?

Short Answer

Expert verified
Answer: The approximate absolute pressure in intergalactic space is 4.14 x 10⁻¹⁸ Pa.

Step by step solution

01

Data given in the question

We are given the following data: - Average density of hydrogen atoms: 1 atom/\(\mathrm{cm}^{3}\) - Temperature: \(3 \mathrm{K}\) We will use the ideal gas law to calculate the absolute pressure.
02

Ideal gas law and constants

The ideal gas law can be expressed as: \(P V = n R T\) where \(P\) - Pressure (in Pascals) \(V\) - Volume (in cubic meters) \(n\) - number of moles of gas \(R\) - Ideal gas constant (\(8.314 \mathrm{J\, mol^{-1} K^{-1}}\)) \(T\) - Temperature (in Kelvin) Also, we need to recall: - Avogadro's number: \(N_A = 6.022 \times 10^{23}\, \mathrm{atoms/mol}\) - Boltzmann's constant: \(k_B = 1.38 \times 10^{-23}\, \mathrm{J/K}\) The pressure we want to find is: \(P = \frac{n R T}{V} = \frac{N k_B T}{V' * V}\) where \(N\) - number of hydrogen atoms per unit volume \(V'\) - Volume of unit hydrogen atom \(V\) - total volume of the space
03

Number of hydrogen atoms per cm³

We are given that there is one hydrogen atom per cm³. Hence, the number of hydrogen atoms per unit volume is: \(N = \frac{1\, \mathrm{atom}}{\mathrm{cm}^{3}}\).
04

Convert volume to meters³

It's important to note that all units in the formula should be in SI (International System) units. So, we need to convert volume from cubic centimeters to cubic meters: \(V' = \frac{1\, \mathrm{cm}^3}{10^6} = 10^{-6}\, \mathrm{m}^3\)
05

Apply the formula for pressure

Now we can use the ideal gas law to calculate the absolute pressure in intergalactic space: \(P = \frac{N k_B T}{V' * V} = \frac{1\, \mathrm{atom/cm^3} \cdot 1.38 \times 10^{-23}\, \mathrm{J/K} \cdot 3 \mathrm{K}}{10^{-6}\, \mathrm{m^3}}\) \(P = 4.14 \times 10^{-18} \, \mathrm{Pa}\) In intergalactic space, the absolute pressure is approximately \(4.14 \times 10^{-18}\, \mathrm{Pa}\).

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Most popular questions from this chapter

A diver rises quickly to the surface from a 5.0 -m depth. If she did not exhale the gas from her lungs before rising, by what factor would her lungs expand? Assume the temperature to be constant and the pressure in the lungs to match the pressure outside the diver's body. The density of seawater is $1.03 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$
Agnes Pockels \((1862-1935)\) was able to determine Avogadro's number using only a few household chemicals, in particular oleic acid, whose formula is \(\mathrm{C}_{18} \mathrm{H}_{34} \mathrm{O}_{2}\) (a) What is the molar mass of this acid? (b) The mass of one drop of oleic acid is \(2.3 \times 10^{-5} \mathrm{g}\) and the volume is $2.6 \times 10^{-5} \mathrm{cm}^{3} .$ How many moles of oleic acid are there in one drop? (c) Now all Pockels needed was to find the number of molecules of oleic acid. Luckily, when oleic acid is spread out on water, it lines up in a layer one molecule thick. If the base of the molecule of oleic acid is a square of side \(d\), the height of the molecule is known to be \(7 d .\) Pockels spread out one drop of oleic acid on some water, and measured the area to be \(70.0 \mathrm{cm}^{2}\) Using the volume and the area of oleic acid, what is \(d ?\) (d) If we assume that this film is one molecule thick, how many molecules of oleic acid are there in the drop? (e) What value does this give you for Avogadro's number?
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. \(]\)
The inner tube of a Pyrex glass mercury thermometer has a diameter of $0.120 \mathrm{mm} .\( The bulb at the bottom of the thermometer contains \)0.200 \mathrm{cm}^{3}$ of mercury. How far will the thread of mercury move for a change of \(1.00^{\circ} \mathrm{C} ?\) Remember to take into account the expansion of the glass. (W) tutorial thermometer)
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