The Air In A Room With Volume 180 M³ Contains 0.35 Carbon Di
The Air In A Room With Volume180 M3contains035carbon Dioxide Initial
The problem involves analyzing the concentration of carbon dioxide in a room over time, given initial conditions, inflow and outflow rates, and the composition of incoming air. The goal is to derive a function p(t) that describes the percentage of carbon dioxide in the room as a function of time t (in minutes) and to determine its long-term behavior.
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The scenario describes a room with a volume of 180 cubic meters (m³) where the initial concentration of carbon dioxide (CO₂) is 0.35%. Fresh air containing 0.05% CO₂ flows into the room at a rate of 2 m³/min. Simultaneously, the mixed air from the room flows out at the same rate of 2 m³/min. The primary objective is to find the function p(t), which represents the percentage of CO₂ in the room over time, and to analyze its behavior as t approaches infinity.
To approach this problem, we adopt a mass balance perspective, considering the amount of CO₂ in the room and how it changes over time. Let P(t) denote the amount of CO₂ in the room at time t measured in cubic meters of CO₂. Since the volume of the room remains constant at 180 m³, and the concentration of CO₂ at any time t is given by p(t), expressed as a percentage, then the actual amount of CO₂ is:
P(t) = (p(t)/100) × V
where V = 180 m³ is the volume of air in the room.
Initial amount of CO₂ in the room is:
P(0) = (0.35/100) × 180 = 0.0035 × 180 = 0.63 m³
Because the flow rates of air into and out of the room are the same, the volume remains constant, and the rate of change of CO₂, dP/dt, depends on the inflow of CO₂ and the outflow of CO₂ from the room. The incoming air has a CO₂ concentration of 0.05%, thus the inflow of CO₂ per unit time is:
Inflow CO₂ rate = flow rate × concentration of incoming air = 2 m³/min × 0.0005 = 0.001 m³/min
The outflow carries away CO₂ at a rate proportional to the current concentration in the room:
Outflow CO₂ rate = flow rate × current CO₂ concentration = 2 m³/min × (p(t)/100) = 2 × p(t)/100
Expressed in terms of P(t), this outflow rate is:
Outflow CO₂ rate = 2 × (P(t)/V) = 2 × (p(t)/100)
The differential equation governing the CO₂ amount P(t) in the room is thus:
\[
\frac{dP}{dt} = \text{Inflow of CO₂} - \text{Outflow of CO₂} = 0.001 - 2 \times \frac{P(t)}{180}
\]
Simplify the outflow term:
\[
\frac{dP}{dt} = 0.001 - \frac{2}{180} P(t) = 0.001 - \frac{1}{90} P(t)
\]
This is a first-order linear differential equation:
\[
\frac{dP}{dt} + \frac{1}{90} P(t) = 0.001
\]
To solve, determine the integrating factor:
\[
\mu(t) = e^{\int \frac{1}{90} dt} = e^{t/90}
\]
Multiplying through by the integrating factor:
\[
e^{t/90} \frac{dP}{dt} + e^{t/90} \frac{1}{90} P(t) = 0.001 e^{t/90}
\]
This simplifies to:
\[
\frac{d}{dt} \left( P(t) e^{t/90} \right) = 0.001 e^{t/90}
\]
Integrate both sides:
\[
P(t) e^{t/90} = \int 0.001 e^{t/90} dt + C
\]
The integral:
\[
\int e^{t/90} dt = 90 e^{t/90} + constant
\]
Thus:
\[
P(t) e^{t/90} = 0.001 \times 90 e^{t/90} + C = 0.09 e^{t/90} + C
\]
Solving for P(t):
\[
P(t) = e^{-t/90} \left( 0.09 e^{t/90} + C \right) = 0.09 + C e^{-t/90}
\]
Apply initial conditions at t=0, where P(0) = 0.63 m³:
\[
0.63 = 0.09 + C e^{0} \Rightarrow C = 0.63 - 0.09 = 0.54
\]
Therefore, the explicit function for the amount of CO₂ in the room is:
\[
P(t) = 0.09 + 0.54 e^{-t/90}
\]
Expressed as a percentage of the room's air, p(t), is:
\[
p(t) = \frac{P(t)}{V} \times 100 = \frac{0.09 + 0.54 e^{-t/90}}{180} \times 100
\]
Calculating numerically:
\[
p(t) = \left( \frac{0.09}{180} + \frac{0.54}{180} e^{-t/90} \right) \times 100 = (0.0005 + 0.003 e^{-t/90}) \times 100
\]
\[
p(t) = 0.05 + 0.3 e^{-t/90}
\]
This is the function describing the percentage of CO₂ in the room over time.
As t approaches infinity, the exponential term e^{-t/90} approaches zero, and the percentage of CO₂ approaches the equilibrium value:
\[
\lim_{t \to \infty} p(t) = 0.05 + 0.3 \times 0 = 0.05\%
\]
Thus, in the long run, the CO₂ concentration stabilizes at 0.05%, reflecting the inflow composition of fresh air that continuously dilutes the room's air.
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