Answer:
• 1.62432 moles of nitrogen
• Tire Pressure: 2.74 * 10⁵ Pa
• The tires will burst
• Pressure: 244 kPa
Explanation:
• We can determine the number of moles of nitrogen using the formula pV = nRT, where p = pressure, V = volume, n = number of moles, R = gas constant, and T = absolute temperature.
Now remember we have our initial pressure in kilopascals so let's convert to pascals (249 pascals). The volume is given in liters, so let's convert into m². And the initial temperature is given in Celsius ⇒ our absolute temperature in Kelvins.
[tex]\mathrm{p\:}=\mathrm{249 kPa\:} = \mathrm{2.49 * 10^5\:},\\\mathrm{15.6L\:} =\mathrm{0.0156m^2\:},\\\mathrm{R\:}=\mathrm{8.314J/mol*K\:},\\\mathrm{T\:}=\mathrm{21C\:} + \mathrm{273\:}=\mathrm{294K\:}[/tex]
Respectively the moles of nitrogen in each tire should be:
[tex]\mathrm{n\:}=\mathrm{pV/RT\:}=\mathrm{(2.49*10^5)(0.0156)/(8.314)(294)\:}=\frac{\left(2.49\cdot \:10^5\right)\left(0.0156\right)}{\left(8.134\right)\left(294\right)}=\frac{3884.4}{2391.396}\\[/tex]
[tex]= 1.62432\dots \mathrm{moles\:}\mathrm{of\:}\mathrm{nitrogen\:}[/tex]
• We can solve this part similarly. All our values will be the same, besides the temperature, as we have to consider both the initial and final temperature here.
[tex]\mathrm{T_2\:}=\mathrm{51C+ 273\:} }=\mathrm{324K\:} }[/tex] -
[tex]\mathrm{p_2\:}=\mathrm{(2.49*10^5)(324)/(294)\:} }=\frac{\left(2.49\cdot \:10^5\right)\left(324\right)}{294}=\frac{40338000}{147}=274408.16326\dots[/tex]
[tex]=2.74408.16326*10^5\dots\mathrm{Pa}[/tex]
• The text mentions that the tires will burst when the internal pressure reaches 269kP. From part #2 we know that the final pressure will be, in kilopascals, 274kP. As 274 > 269, the tires will burst in Death Valley.
• We would want the final temperature = breaking pressure. Therefore,
[tex]\mathrm{p_2\:}=\mathrm{(269)(294)/(324)\:} }=\frac{79086}{324}=\frac{13181}{54}=244.09259\dots\mathrm{kPa\:} }[/tex]
