Answer:
x(t) = −39e
−0.03t + 40.
Explanation:
Let V (t) be the volume of solution (water and
nitric acid) measured in liters after t minutes. Let x(t) be the volume of nitric acid
measured in liters after t minutes, and let c(t) be the concentration (by volume) of
nitric acid in solution after t minutes.
The volume of solution V (t) doesn’t change over time since the inflow and outflow
of solution is equal. Thus V = 200 L. The concentration of nitric acid c(t) is
c(t) = x(t)
V (t)
=
x(t)
200
.
We model this problem as
dx
dt = I(t) − O(t),
where I(t) is the input rate of nitric acid and O(t) is the output rate of nitric acid,
both measured in liters of nitric acid per minute. The input rate is
I(t) = 6 Lsol.
1 min
·
20 Lnit.
100 Lsol.
=
120 Lnit.
100 min
= 1.2 Lnit./min.
The output rate is
O(t) = (6 Lsol./min)c(t) = 6 Lsol.
1 min
·
x(t) Lnit.
200 Lsol.
=
3x(t) Lnit.
100 min
= 0.03 x(t) Lnit./min.
The equation is then
dx
dt = 1.2 − 0.03x,
or
dx
dt + 0.03x = 1.2, (1)
which is a linear equation. The initial condition condition is found in the following
way:
c(0) = 0.5% = 5 Lnit.
1000 Lsol.
=
x(0) Lnit.
200 Lsol.
.
Thus x(0) = 1.
In Eq. (1) we let P(t) = 0.03 and Q(t) = 1.2. The integrating factor for Eq. (1) is
µ(t) = exp Z
P(t) dt
= exp
0.03 Z
dt
= e
0.03t
.
The solution is
x(t) = 1
µ(t)
Z
µ(t)Q(t) dt + C
= Ce−0.03t + 1.2e
−0.03t
Z
e
0.03t
dt
= Ce−0.03t +
1.2
0.03
e
−0.03t
e
0.03t
= Ce−0.03t +
1.2
0.03
= Ce−0.03t + 40.
The constant is found using x(t) = 1:
x(0) = Ce−0.03(0) + 40 = C + 40 = 1.
Thus C = −39, and the solution is
x(t) = −39e
−0.03t + 40.
<u>Answer:</u> The final temperature of the copper is 95°C.
<u>Explanation:</u>
To calculate the final temperature for the given amount of heat absorbed, we use the equation:
Q = heat absorbed = +133 J (heat is added to the system)
m = mass of copper = 5.00 g
c = specific heat capacity of copper = 0.38 J/g ° C
Putting values in above equation, we get:
Hence, the final temperature of the copper is 95°C.
Answer:
lower
Explanation:
The higher the sea level, the lower the atmospheric pressure. This is due to the density of air decreasing as the altitude increases.
Answer:
n l m
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1 0 0 1s 1 2 2
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2 0 0 2s 1 2
2 1 1,0,-1 2p 3 6 8
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3 0 0 3s 1 2
3 1 1,0,-1 3p 3 6
3 2 2,1,0,-1,-2 3d 5 10 18
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4 0 0 4s 1 2
4 1 1,0,-1 4p 3 6
4 2 2,1,0,-1,-2 4d 5 10
4 3 3,2,1,0,-1,-2,-3 4f 7 14 32
Explanation:
