Question 282: Units , Dimensions and Error Analysis

Question

Pressure depends on distance as, $$P = {\alpha  \over \beta }\exp \left( { - {{\alpha z} \over {k	heta }}} ight)$$, where $$\alpha $$, $$\beta $$ are constants, z in distance, k is Boltzman's constant and $$	heta $$ is temperature. The dimention of  $$\beta $$ are

Accepted Answer

The given expression is
$$P = {\alpha  \over \beta }\exp \left( { - {{\alpha z} \over {k	heta }}} ight)$$
Now,
$$[z] = [{L^1}]$$
$$[	heta ] = [{K^1}]$$
$$[k] = [{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}]$$
Since the terms which are raised to power in an exponential function are dimensionless, we get
$$[\alpha] = \left[ {{{k	heta } \over z}} ight] = [{M^1}{L^1}{T^{ - 2}}]$$
Also, $$[P] =$$ $$\left[ {{\alpha  \over \beta }} ight]$$
$$ \Rightarrow $$ $$\left[ \beta  ight] = \left[ {{\alpha  \over P}} ight]$$
$$ \Rightarrow $$ $$\left[ \beta  ight] = \left[ {{{{M^1}{L^1}{T^{ - 2}}} \over {M{L^{ - 1}}{T^{ - 2}}}}} ight]$$
= $$[{M^0}{L^2}{T^0}]$$
Answer (C)

Answer

[M
0
L
0
T
0
]

Answer

[M
-1
L
-1
T
-1
]

Answer

[M
0
L
2
T
0
]

Answer

[M
-1
L
1
T
2
]

Units , Dimensions and Error Analysis

Pressure depends on distance as, $P = {\alpha \over \beta }\exp \left( { - {{\alpha z} \over {k\theta }}} \right), where \alpha , \beta are constants, z in distance, k is Boltzman's constant and \theta is temperature. The dimention of \beta $ are

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