where h is the Planck constant and p is the momentum. (In this case, we need to use the version h = 4.13 x 10-15 eV*s)
Since the problem information provides an energy E = 1 MeV, then we need p in the form:
p = [2mE]1/2
These units mean it will also be easier to do the problem by using atomic mass units in MeV. Thus, 1 u = 931 MeV/c2, so for an alpha particle (m =He4) we need 4u = 4 (931 MeV/c2)
Then:
l D = (4.13 x 10-`5 eV*s)/ 2[4 (931 MeV/c2)] = 10-14 m
l D = 10-12 cm
32) For a Young double slit, d sin (q) = m l
gives the bright fringe, where m is the order and l the wavelength. Since the distance to the screen is much greater than the height of the bright fringe, sin (q) = x / D, where x is the height and D is the distance to the screen. So:
D (x) d/ D = l
D(x) = Dl/ d
then: D(x) = (3 x 103 mm) (6 x 10-5 cm/ 0. 2 cm) = 0.9 mm
33) This application makes use of the energy-time uncertainty principle:
D(E) D(t) = h/ 2 π
Here, t = the mean lifetime of the excited state = T(½) = 8 x 10-8 sec
And E= the half-width of the excited state = E(½)
then:
E(½) T(½) =h/ 2 π = 1.03 x 10-34 J-s
E(½) = (1.03 x 10-34 J-s)/ (8 x 10-8 sec) = 1.2 x 10-27 J
But: 1.6 x 10-19 J = 1 eV (electron -volt)
so: E(½) = 10-9 eV
34) We have: N = No exp (-lt)
let t = T1/2 the half life, then:
N = No exp (-lT1/2)
or N / No = ½ = exp (-lT1/2)
and: - ln 2 = (-lT1/2)
T1/2 = (ln 2)/ l
35) The "mean life" of one of the atoms in the sample will be:
since exp(0) = 1
36) We need to apply both conservation of energy, and conservation of momentum. (There are two distinct angles involved, φ and q)
For the conservation of energy: T1 = T1' + T2'
for the conservation of momentum : P1 = P1'cos(q) + P2' cos (φ)
and: 0 = P1'sin(q) - P2' cos(φ)
p = [2mT]1/2 (see also #31)
A head-on collision implies the unique values: φ = 0 and q = 180 deg
so:
[2m1T1]1/2 = ([2m1T1']1/2 ) cos(180) + ([2m2T2']1/2 ) cos (0)
But cos (0)= 1 and cos (180) = -1, so:
[2m1T1]1/2 = - [2m1T1']1/2 + [2m2T2']1/2
Now, since T1 = T1' + T2':
2m1T1 + 4m1 [T1T1']1/2 = 2m2T2' - 2m1T1'
4m1 [T1T1']1/2 = - 2m1T1' + 2m2T2' - 2m1(T1 - T2')
whence:
2m1[T1(T1 - T2')]1/2 = (m1 + m2)T2' - 2m1 T1
4m12 T1 (T1 - T2') = (m1 + m2)2T2' + 4m12T12 - 4m1T1(m1 + m2)T2'
(m1 + m2)2 T2'2 - 4 m1m2T1T2' = 0
Then:
T2/T1 = 4 m1m2/ (m1 + m2)2
37) We sketch the HCl molecule, so:
H o<-------- cm-------------------="" x="">O (Cl)
And 1.27 Å defines the entire distance as shown (arrow to arrow) and let 'x' be the distance from the center of mass (cm) to the H atom. (Recall 1Å = 10-8 cm).
Then:
x = 36(1.27Å - x)
37x = 36 (1.27 Å)
x = 36(1.27 x 10-8 cm)/ 37
m(p) = 1.67 x 10-24 g (mass of proton)
The moment of inertia I is given by:
I = m(p)2 x2 + 36 m(p) [(1.27 x 10-8 cm) - x]2
I = (1.67 x 10-24 g)(36/37)2 + 36/(37)2 [(1.27 x 10-8 cm)]2
I = 2.6 x 10-40 g-cm^2
38) The Lande g-factor is defined:
g = [ J(J+ 1) + S(S +1) - L(L +1)/ 2J(J + 1) ] + 1
From our previous exposure to quantum mechanics
See, e.g.
http://brane-space.blogspot.com/2010/07/l-s-coupling-problem-solutions-and-more.html
and:
http://brane-space.blogspot.com/2010/07/more-quantum-problems.html
the 1D 3/2 state implies: L = 2, S = 1 and J = L + S = 2 + 1 = 3
then:
g = [3(3 + 1) + 1(1 + 1) - 2( 2+ 1)/ 2(3)(3 + 1)] + 1 = 1.33
39) Recall the nuclear relation:
f = M(A, Z) - A'/ A
and that graphing this vs. A yields a constituent mass 1.0085A. Thus the total binding energy is: (0.0085A) x (931 MeV) or about 7.9 MeV per nucleon.
40) Here, it's important to recognize 2144 cm-1 as the wave number, k. Also:
k = f/c = 2144 cm-1
then the frequency:
f = kc = (2144 cm-1) (3 x 1010 cm/sec) = 6.43 x 1013/s
41) Since the current decreases uniformly, we don't need di/dt and can instead make use of the finite difference: D(I)/ D(t)
And we know the voltage:
Therefore: V = 8 volts
42) Earth's atmosphere extends about 50 miles. If we assume the density of air to be constant:
p2 - p1 = r g(z2 - z1)
where p2, p1 are the different pressures at heights z2 and z1, respectively, and r is the air density with g the acceleration of gravity.
We can form a simple proportion based on the simplifying assumption:
(p2 - p1)/ p2 = (z2 - z1)/z2
p2 is what we need to find, and z2 is the height for 50 miles or ~ 50 x 5,000' = 250,000'.
whence:
(z2 - z1) = 200'
(p2 - p1)/ p2 = (z2 - z1)/z2 = 200'/ 250,000' = 0.0008
so:
(p2 - p1) = 0.0008 (p2)
p1 = p2 - 0.0008 (p2) = p2 = 76 Hg
43) For this solution, we let all quantities inside the sphere be denoted by (1) and all the quantities outside by (2).
By Gauss' law:
4 π r2 E1 = 4 π r(4 πr2/ 3)
where r (charge density) = 3 q / (4 π a2)
Therefore, E1 = qr/ a3
Further the associated energy is obtained via integration:
W1 = 1/ 8 π [∫ (0 to a) E12 (dA)
= 1/ 8 π [∫ (0 to a) (qr/ a3)2 (4 π r2) dr
W1= q2/ 10a
By Gauss' law:
4 π r2 E = 4 πq
E2 = q/ r2
W2 = 1/ 8 π [∫ (a to oo) (q/ r2)2 (4 π r2) dr
W2 = q2/2 [∫ (a to oo) dr/ r2]
W2 = q2/ 2a
Therefore: the total electrostatic energy W = W1 + W2
W = q2/ 10a + q2/ 2a = 3q2/ 5a
44) It helps here to sketch the circuit ( see graphic) and make use of angular frequency, w
From this, the total impedance is (bear in mind j = [-1]-1/2:ZT = 1/ j wL + 1/ (1/ jw C1) + 1/ (1/ jwC2)
ZT = 1/ jwL + jw( C1 + C2)
ZT = j[ - 1/wL + w(C1 + C2)]
ZT = -1/wL + w(C1 + C2) = 0 at the natural frequency
so:
w2 = 1/ L(C1 + C2)
w = [1/ (10-5) (30 x 10-6)]1/2
w = 105 [1/3]1/2
w = 57,735 /s = 2 π f
so the natural frequency f = w/ 2 π = 9, 189 /s
or about 9.2 kilocycles per second
45) From thermodynamic relations we have:
C V dT = C p - C v / (¶V/ ¶T) V dV
Then:
(¶T/¶V)s= (C p / C v - 1) (¶T/¶V) p
But:
(¶T/ ¶V) p = T/V since T = PV/ nR
dT/ T + (C p/ C v - 1) (dV/V) = 0
Integrating both sides:
TV^ (C p/ C v - 1) = k
PV^ (C p / C v ) / nR = k
PV^(C p/ C v ) = k/ nR
Then:
n = k/ PV^[C p/ C v ] R
No comments:
Post a Comment