3. Is Neutron Star Matter Dangerous? Suppose you have the technology to teleport
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3. Is Neutron Star Matter Dangerous? Suppose you have the technology to teleport one teaspoon of matter from inside a neutron star into your lab on Earth. Would this be a good idea? Let's find out: a. Explosive Decompression. Assume that at the in- stant the neutron star matter arrives in your lab it has nuclear density (pnucl2x1017 kg/m3). Also let's suppose your lab room is a cube of side length 10 m. expands. Thus, energy conservation tells us that the total energy of the gas does not change as it expands. Since the total energy of a gas is the sum of the ki- netic energies of its constituent particles, the kinetic energies of the constituent particles does not change, either. In other words, the kinetic energy of each neutron, KE--kT remains constant during the expansion. Estimate the total kinetic energy of all the neutrons in your expanded sample. This is called the thermal energy of the gas. If you doused the contents of your lab room in the Arctic Ocean, estimate by how many degrees the temperature of the Arctic Ocean would rise. (The volume of the Arctic Ocean is about 19 million km8, and it takes about 4,200 J to heat one kg of water by one degree Celsius). Instant global warming! the weight of Mount Everest (~ 1.6X1015 N). Do you think the walls of your lab could contain matter at this pressure? i. About how many neutrons are in your tea- spoon-sized sample (1 tsp 5 mL)? ii. As a gas originally under great pressure your teaspoon of neutron star matter will expand very quickly to fill your lab room. The pressures in volved are so great that you can completely neglect the air in the room-it might as well be a vacuum. What is the density of the gas when it has filled your lab room? How many times denser than lead is this gas? iv. Using the same formula, KEkT, esti mate the typical speed of the neutrons, and hence the expansion velocity. Compare this expansion ve locity to the detonation velocity of TNT (6,940 m/s). Estimate how long it will take the gas to expand to fill the lab room. This is thousands of times faster than an air bag in a car takes to inflate iii. When a neutron star first forms it has a hot surface, on the order of a million K, and an even hot ter interior. Suppose your sample (from inside the star) is initially at a temperature of 10 million K. Since your sample is basically expanding into a vacuum (see comment in part (ii), it does negligible work as it expands. Moreover, there is no thermal energy ("heat") being added or removed from the gas as it v. Using the ideal gas law, estimate the pres sure of the gas when it has expanded to l the lab room. How much force does this pressure exert orn one of the walls of the lab room? Compare this toExplanation / Answer
i) density of neutron star matter = 2.0e+17kg/cu.m
volume of the sample = 5ml = 5.0e-6 cu.m
mass of sample = 5.0e-6 *2.0e+17 = 1.0e+12 kg
mass of neutron = 1.673e-27 kg
number of neutrons in the sample = 1.0e+12/ 1.673e-27 = 5.98 e+38
volume of the gas when completely filled the lab room = 103 cu.m
density of the gas when completely filled the lab = 1.0e+12 /1.0e+3 = 1.0e+9
The gas is denser than lead by an order of 1.0 e+8
iii) temp. of the neutron gas = 1.0e+7 K
KE of the neutron = 3kT/2 = 3/2 *1.0e+7 * 1.38e-23
= 2.07e-17 J
Total KE of the sample = 2.07e-17 *5.98e+38
= 1.24e+22 J
volume of the Ocean = 19.0e+6 km3 = 19.0e+15 cu.m
density of water = 1000kg/cu.m
mass of water in the ocean = 19.0e+15*1000 = 19.0e+18 kg
heat required 4200J/kg/degcel
tmep. raise by dousing the neutron mass = 1.24e+22 /( 19.0e+18*4200)
= 0.15 C
iv) KE of neutron = 2.07 e-17 J
mass of neutron = 1.67e-27 kg
speed of neutron = v= (2*2.07e-17/1.67e-27)1/2
= 1.57e+5 m/s
iv) average time for a neutron to reach any wall of the room = 10/1.57e+5
= 6.35e-6 s
typiclly within a micro second the gas fills the room
v) PV = nRT
mass of the gas = 1.0e+12 kg
number of moles n = 1.0e+15/1.0 = 1.0e+15 ( typically molecular wieght of H-2 is 1 gm)
P = nRT/V = 1.0e+15 *8.314*1.0e+7/1.0e+3
= 8.314 e+19 Pa
Force on the walls = PA = 8.314e+18*1.0e+2 = 8.314e+20 N
This is a millison times the weight of Mount everest.
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