Linear relations-examples
1. In general, the respiratory rate r(in breaths/min) of an individual is linearly related to the partial pressure P CO2 (in torr: 1torr=1mm
Hg; 760torr=1atmosphere) of carbon dioxide in the lungs. (partial pressure of CO 2 = total gas pressure § proportion of CO2 in the
gas mixture). [In the lungs, total pressure at rest (at sea level) is about 760 torr; the gas mixture in the lungs consists of water vapor,
air (nitrogen, oxygen, carbon dioxide, etc.), and waste products -including CO2 - extracted from the blood]. In an experiment, an
individual inhales air from a bag containing approximately 2 percent carbon dioxide, producing a partial pressure P CO2 of 41 torr
and a corresponding respiratory rate r of 13.8 breaths/min; subsequently, the same individual inhales air containing approximately 6
percent carbon dioxide, giving a PCO2 of 50 torr and a respiratory rate r of 19.1 breaths/min. We will find the specific linear
relation giving r in terms of PCO2 for this individual, and use this relation to calculate the respiratory rate when the partial pressure
of carbon dioxide is 45 torr and the rate at which respiratory rate changes.
The data given can be considered two pairs of the form ( P CO2 , r ): (41,13.8) and (50,19.1). (with r second because we want r
as dependent variable) The relation is linear, so we compute slope (rate of change of r with respect to P CO2 ):
r
r(end) - r(start)
19.1 - 13.8
breaths/min
m =
=
= 50 - 41 = .59
torr
PCO2 PCO2(end) - PCO2(start)
and we can find the equation by: ( r - 13.8) = .59( PCO2 -41), so that r = -10.39 + .59PCO2. To obtain the respiratory rate when P CO2
is 45 torr we use this equation, and find
r = -10.39 + .59(45) = 16.16 breaths/min. The rate of change of the respiratory rate with respect to P CO2 is given by the slope; it
is .59 breaths/min/torr
2. Measurements of temperature on the three common scales (Celsius, Fahrenheit, and Kelvin) are linearly related. [The Kelvin scale is
often called the absolute scale because of the nicer form of many physical laws in °K and because 0°K is the temperature at which
molecules have no kinetic energy-no heat at all.] The temperature that gives a reading of 0 ° on the Celsius scale (the temperature at
which pure water forms ice at pressure 1torr) gives a reading of 32 °F and 273.18°K (usually rounded to 273°K except in quite
precise work), and the temperature which gives a reading 100 °C gives readings of 212°F and 373.18°K . To express the Fahrenheit
reading in terms of the Celsius reading, we set F = Fahrenheit reading, C = Celsius reading, and note that we have two pairs ( C , F
) ( F second, because we wish it to be dependent) (0, 32) and (100, 212). We can compute
F
212 - 32
9
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slope:
= 100 - 0 = 5 and use (F - 32) = 5 (C - 0) to obtain the equation:
C
9
9
F = 5 C + 32 The slope ( 5 ) gives the rate of change of Fahrenheit temperature with respect to Celsius temperature - For each
9
one degree increase in Celsius temperature, the Fahrenheit reading increases by 5 degrees.
3. In females of the snake Lampropeltis polyzona the total length y is linearly related to the tail length x for tail lengths between
30mm to 200mm and total lengths from 200mm to 1400mm. In one specimen, the tail length was 60mm and the total length 455mm,
in another, the tail length was 140mm and the total length 1050mm.. To give the function expressing total length in terms of tail
length, we note the data give us
two pairs ( x , y ) ( y is to be dependent): (60, 455) and (140,1050), so that the slope will
y 1050 - 455
be
= 140 - 60 = 7.44, and the equation comes from ( y -1050) = 7.44(x-140) so we obtain y = 8.4 + 7.44x. The slope 7.44
x
gives the rate of change of total length with respect to tail length (7.44 inches total length added for each added inch of tail)
4. In respiratory physiology, the volume of a particular breath is called the tidal volume, denoted VT , and is expressed as a fraction of
the vital capacity of the individual (Vital capacity is the maximum amount of air that can be exhaled after maximum possible
inhalation - it is the usable volume of the lungs, smaller than the total volume). During regular breathing, the time period t (in sec.)
of the breathing cycle is linearly related to the tidal volume of the breath, with the relation given by t = 0.14 + 1.5 VT .
For example, a person whose vital capacity is 5 liter, breathing 1.6 liter per breath, moves a tidal volume of .32, and this model
predicts a time period for the breath of .14 + 1.5(.32) = .62 sec. The slope 1.5 gives the rate of change of the length of a breath with
respect to tidal volume: a change of 1 unit in tidal volume (if it were possible) would change the length of the breath by 1.5 sec.
(More realistically, a change of .01 unit would change the length of a breath by .015 sec.)
5. A very important special case of linear relation is proportion, in which the intercept is 0, so that y = kx for some constant k (this is
also often expressed as y/x = k - "y and x have the same relative size as they change" - and the notation used to indicate
proportionality is y å x. The constant k is called the constant of proportionality : it can often be interpreted as a rate, but tends to
have strange units of measurement.). In a proportional relationship, when the value of one variable doubles, so does the other; if one
value is multiplied by 1.5, so is the other, etc. - the percentage change is the same in both.
For example, consider some special cases of Boyle's law relating temperature (in°K), pressure (in torr) , and volume (in liters) for n
moles of a fixed quantity of a gas:
62.4n
P = V T, with P=pressure(torr), V=volume(l), T=temperature( °K) [any units for pressure and volume will work - though the
value 62.4 would change - but using either Celsius or Fahrenheit scales for temperature destroys the proportionality because the
meaning of 0 changes].
If a fixed quantity of a gas is kept at a fixed volume (in a rigid containment vessel, for example), the pressure exerted (torr) will be
proportional to the temperature of the gas in degrees Kelvin; that is, we can write P=C1T, using C1 to represent the constant of
proportionality (which will depend on the amount of the gas and the volume). If a quantity of gas is held at a fixed volume and exerts
a pressure of 760 torr(balancing atmospheric pressure) at a temperature of 300°K, then C1 for this sample is 2.53, so P = 2.53T; then
at 273°K (the freezing point of water) the pressure exerted on the containment vessel would be about 691 torr (about 13.4 pounds
per square inch). If the temperature is doubled (doubling temperature only makes sense with °K), to 546°K, the pressure will also
double, since we have a proportion.
On the other hand, if our fixed quantity of gas is maintained at the same pressure, the volume occupied will be proportional to
the temperature in °K and we obtain the law of Charles and Gay-Lussac V = C2T (C2 depending on the amount of gas and the
pressure).
If temperature is measured in °F or °C, the relationship between temperature and pressure is still linear, but is not a proportion.
Linear relations-exercises
1.) A study in 1971 in Oslo, Norway, suggests a linear relation between the concentration C (in µg/m3) of SO2 in the air and the
number of deaths N during the week in the city, given by N = 94 + .031C when C is in the range 50 µg/m3 < C < 700 µg/m3.
a.) How many deaths would this model predict in a week in which the concentration of SO 2 is 150 µg/m3?
b.) What is the rate of change of the number of deaths (per week) with respect to the concentration of SO 2 ?
c.) If the mean concentration C increases by 200 µg/m3 from one week to the next, what change in the number of deaths would the
model predict [Note that you don't need to know the actual concentration or the number of deaths - just the change]?
2.) The vital capacity of a normal healthy young man is 5 liter. During one period, he breathes so that the volume of each breath is 1.8
liter. Calculate the time period of each breath. If his breathing changes so that the volume of each breath is 2.0 liter, what change in
the time period of each breath will occur?
3.) The vital capacity of a young woman is 3.5 liter. During one period, the time of each breath is .68 seconds. Calculate the volume of
each breath.
4.) If the pressure exerted by a sample of gas is .75 torr at 0°C, what is the pressure at 25°C if the volume is kept constant? What is its
pressure at -20°C at the same volume? (Note that you need to either convert the temperature measurements to °K or find the formula
for pressure in terms of °C) What is the rate of change of pressure with respect to temperature for this sample?
5.) When partial pressure of CO2 and respiratory rate are compared for one individual, it is found that for a partial pressure of CO 2
equal to 41torr the rate is 14.2 breaths/min. and for a PCO2 equal to 50 torr the respiratory rate is 18.9 breaths/min.
a.) Give the formula expressing respiratory rate in terms of P CO2 for this individual and
b.) Find the respiratory rate for a P CO2 equal to 45 torr.
c.) At what value of PCO2 will he respiratory rate be 17 breaths/min.?
6.) By substituting TK = TC + 273 (TK = temperature reading in °K, TC = temperature reading in °C), show that the law of Charles and
V0
Gay-Lussac can be rewritten as V = V0 + 273 TC, where V0=volume of the sample at temperature 0°C and TC is temperature
measured in °C.
7.) If a sample of gas has volume 5 liter at temperature 0 °C, what will be the volume of the sample under the same pressure at
temperature 25°C? At what temperature will the volume reach 7.5 liter?
8.) Give the formula expressing temperature in °C in terms of temperature in °F. Normal (oral) temperature in humans is sbout 98.6°F;
what is this in °C?
9.) The volume y (in milliliters) of carbon dioxide dissolved per 100 milliliters of arterial blood is expressed as a function of the partial
pressure of carbon dioxide PCO2 (in torr) by y = .0618 P CO2 . For a normal person resting at sea level the partial pressure of CO 2 in
the arterial blood is 46.8 torr. How much CO2 will be dissolved in each 100 ml of arterial bloosd? What is the rate of change of the
volume of dissolved Co2 (per 100 ml. of blood) with respect to partial pressure?
10.) (Refer to #9, above)If the partial pressure of Co 2 dissolved in a quantity of arterial blood decreases from 80 torr to 50 torr (the
person has been diving, under the pressure of the water, and comes up near the surface) what will happen to the amount of CO 2
dissolved per 100 ml of blood?